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                    UNIVERSIDADE FEDERAL DE ALAGOAS
INSTITUTO DE MATEMÁTICA
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA UFAL-UFBA

JOSÉ IVAN DA SILVA SANTOS

SOME GEOMETRIC AND ANALYTICAL RESULTS ON WEIGHTED
RIEMANNIAN MANIFOLDS

DOCTORAL THESIS

Maceió
2016

JOSÉ IVAN DA SILVA SANTOS

SOME GEOMETRIC AND ANALYTICAL RESULTS ON WEIGHTED RIEMANNIAN
MANIFOLDS

Thesis presented to the Post-graduate Program in Mathematics at Instituto de
Matemática da Universidade Federal de
Alagoas as partial fulfillment of the requirements for the degree of Doctor in Philosophy
in Mathematics.

Advisor: Prof. PhD. Márcio Henrique B. da Silva

Maceió
2016

Catalogação na fonte
Universidade Federal de Alagoas
Biblioteca Central
Divisão de Tratamento Técnico
Bibliotecário Responsável: Valter dos Santos Andrade
S237s

Santos, José Ivan da Silva.
Some geometric and analytical results on weighted Riemannian manifolds /
José Ivan da Silva Santos. – 2016.
73 f.
Orientador: Márcio Henrique B. da Silva.
Tese (Doutorado em Matemática) – Doutorado Interinstitucional UFBA/UFAL.
Universidade Federal de Alagoas. Instituto de Matemática. Programa de
Pós-Graduação em Matemática. Maceió, 2016.
Bibliografia: f. 70-73.
1. Variedades Reimannianas ponderadas. 2. Tensor de Bakry-Émery Ricci.
3. Operador de Jacobi ponderado. 4. Estabilidade. 5. Auto valores de Stekloff.
6. Estimativas de auto valores. 7. Teorema splitting ponderado. I. Título.
CDU: 514.764.2

SOME GEOMETRIC AND ANALYTICAL RESULTS ON WEIGHTED RIEMANNIAN
MANIFOLDS

JOSÉ IVAN DA SILVA SANTOS

Thesis presented to the Post-graduate Program in Mathematics at Instituto de
Matemática da Universidade Federal de
Alagoas as partial fulfillment of the requirements for the degree of Doctor in Philosophy
in Mathematics.

Examiners:

For my grandparents, Artur dos Santos and Maria Jovelina, my
parents, Antônio Artur and Djesima Maria, and
my wife Jaaresias Nascimento.

ACKNOWLEGMENTS

Como as pessoas a quem dedico as palavras seguintes têm como primeiro idioma o
português, este será o idioma que escreverei meus agradecimentos.
Primeiramente agradeço a Deus por me conceder saúde e entusiasmo no decorrer de
todo o curso e por trazer todo o necessário para a conclusão desse trabalho.
Sou imensamente grato ao Prof. Dr. Márcio Henrique Batista, orientador desssa tese,
que aceitou o desafio de me conduzir e preparar para o meio cientı́fico. Quero aqui destacar
suas qualidades como profissional e pessoa as quais levarei como exemplo, compromisso,
seriedade e respeito com tudo o que fazemos e com todos que convivemos.
Sou imensamente grato a todos os professores do programa de pós-graduação em
matemática aos quais devo todo o conhecimento aprendido.
Sou grato a todos os colegas do curso de pós-graduação em matemática pela amizade
e contribuições.
Devo meus agradecimentos a CAPES por conceder o apoio finaceiro, sem o qual nada
do que foi feito seria possı́vel.
Agradeço e dedico esta tese a todos os meus familiares que torceram e apoiaram-me
em todos os momentos, fossem eles “fáceis” ou difı́ceis.

ABSTRACT

Let pM, x , y, f q be a weighted Riemannian manifold. In this thesis we obtain some geometric
and analytical results in pM, x , y, f q assuming that Bakry-Emery Ricci tensor is non-negative
in some results in other results we assuming that the weighted mean curvature is bounded
from below. Moreover, assuming that the radial generalized sectional curvature is bounded
from below we obtain a comparison theorem for the Hessian of the distance function and
some consequences of it. Let Σ be a closed surface in M , assuming that the Perelman scalar
curvature is bounded from below, we obtain an upper bound for the first non-zero eigenvalue
of the weighted Jacobi operator for surfaces Σ Ă M and we generalize a result of Shoen and
Yau about stable minimal surfaces, see [45]. We also obtained, for surfaces with boundary,
a sharp estimate from below for the first non-zero Stekloff’s eigenvalue. For surfaces we also
obtain an upper bound for the first non-zero eigenvalue of the weighted Jacobi operator and
some consequences of it, for instance, we show that in R3 there exist no closed stable selfshrinker. In higher dimension we obtain upper bound and lower bound for the first non-zero
Stekloff’s eigenvalue on suitable hypotheses. We conclude our work with a weighted splitting
theorem.

Keywords: Weighted Riemannian manifolds. Bakry-Émery Ricci tensor. Weighted Jacobi
operator. Stability. Stekloff’s eigenvalue. Eigenvalue estimates. weighted splitting theorem.

RESUMO

Seja pM, x , y, f q uma variedade Riemanniana ponderada. Nesta tese obtemos resultados
geométricos e analı́ticos em pM, x , y, f q assumindo que o tensor de Bakry-Émery Ricci é não
negativo em alguns resultados e assumindo que a curvatura média ponderada é limitada
inferiormente em outros. Além disso, assumindo que a curvatura seccional generalizada
radial é limitada inferiormente obtemos um teorema de comparação para a Hessiana da
função distância e algumas consequências. Seja Σ uma superfı́cie fechada em M , assumindo
que a curvatura escalar de Perelman é limitada inferiormente, obtemos um limite superior
para o primeiro autovalor não nulo do operador de Jacobi ponderado da superfı́cie Σ Ă M
e generalizamos um resultado de Schoen e Yau sobre superfı́cies mı́nimas estáveis, veja [45].
Também obtemos, para superfı́cies com fronteira, uma estimativa sharp inferiormente para o
primeiro autovalor não nulo de Stekloff. Para superfı́cies também obtemos um limite superior
para o primeiro autovalor não nulo do operador de Jacobi ponderado e algumas consequências,
por exemplo, mostramos que em R3 não existe self-shrinker fechado e estável. Em dimensão
alta obtemos limites superiores e inferiores para o primeiro autovalor não nulo de Stekloff
sobre hipóteses apropriadas. Concluı́mos nosso trabalho com um teorema splitting.

Palavras chave: Variedades Riemannianas ponderadas. Tensor de Bakry-Émery Ricci.
Operador de Jacobi ponderado. Estabilidade. Autovalores de Stekloff. Estimativas de autovalores. Teorema splitting ponderado.

CONTENTS

1

INTRODUCTION

11

2

PRELIMINARIES

14

3

THE HESSIAN OF THE DISTANCE FUNCTION
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Comparison Theorem to the Hessian of the Distance Function . . .

16
16
17

4

ON THE FIRST STABILITY EIGENVALUE OF SURFACES
27
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Estimates for the First Eigenvalue of the Weighted Jacobi Operator 28
4.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Proof of the Theorems 4.1, 4.3 and 4.4 . . . . . . . . . . . . . . . . . 32
4.4.1 Proof of the Theorem 4.1 . . . . . . . . . . . . . . . . . . . . 32
4.4.2 Proof of the Theorem 4.3 . . . . . . . . . . . . . . . . . . . . 34
4.4.3 Proof of the Theorem 4.4 . . . . . . . . . . . . . . . . . . . . 35

5

STEKLOFF’S EIGENVALUES TO WEIGHTED RIEMANNIAN MANIFOLDS
40
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Weighted Stekloff ’s Eigenvalue Problems . . . . . . . . . . . . . . . 42
5.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Proof of Eigenvalue Estimates and Rigidity . . . . . . . . . . . . . . 46
5.5 Sharp Estimate of the Stekloff ’s Eigenvalue for Surfaces . . . . . . 52

6

A WEIGHTED SPLITTING THEOREM
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Technical Computations . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Proof of the Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . .

9

58
58
59
61
64

BIBLIOGRAPHY

70

CHAPTER 1
INTRODUCTION

The study of Riemannian manifolds endowed with a smooth density function has flourished in last few years, and a much better understanding of their geometrical and analytical structure has evolved. We point out, for instance, the solution of Poincaré conjecture, the relaxation of the conditions for solve the Monge’s problem for mass transportation, the behavior of singularities of the Ricci flow, the mean curvature flow and others, see
[10, 17, 33, 36, 37, 39, 51, 55] and references therein. Moreover, the theory of these spaces
and the generalized curvatures go back to Lichnerowich [31, 32] and more recently by Bakry
and Émery [7], in context of diffusion process, and it has been a very active area in recent
years.
In this thesis we give contributions to the study of weighted Riemannian manifolds. We
obtain some geometric and analytic results on weighted Riemanniann manifolds, and these
generalize some results in [2, 14, 57, 58] for the weighted context. More specifically, inspired in
the work of Impera and Rimold in [28], and using the concept of weighted sectional curvature
defined by Wylie in [53], we obtain a comparison theorem for the weighted Hessian of the
distance function and some consequences to the weighted Laplace-Beltrame operator. In
particular we recover the following estimates to f -Laplacian of the distance function
∆f rpxq ď pm ´ 1q

h1 prpxqq
` θprpxqq,
hprpxqq

obtained in [43], see theorem 3.2. The comparison theorem for the weighted Hessian of the
distance function is the main result of the Chapter 3.
In the Chapter 4, encouraged by ideas in [2, 3, 4, 41], we study some analytical aspects of
surfaces with constant weighted mean curvature. More specifically, we study upper estimates
of the first eigenvalue of the weighted Jacobi operator on closed surfaces. Moreover, we
characterize the equality cases. This study allow us to generalize a result obtained by Schoen
and Yau on stable minimal surfaces in 3-Riemannian manifolds with nonnegative scalar
curvature for the setting of weighted Riemannian manifolds. Some consequences of that
results are obtained, in which we show that all closed λ-surfaces in the Gaussian space are
unstable, in particular, there exist no closed stable self-shrinker surfaces in R3 .
11

In the Chapter 5, motivated by works [14, 57, 58], we study the Stekloff eigenvalue
problems on weighted Riemannian manifolds.
The Classical Stekloff’s eigenvalue problem
#
∆u “ 0
Bu
“ σu
Bν

in Ω,
on BΩ,

was introduced by self in [49] for bounded domains Ω of the plane and later it was studied by
Payne in [42] for bounded domains in the plane with non-negative curvature. This problem
has a physical interest because the eigenfunctions represent the steady state temperature on
a domain and the flux on the boundary is proportional to the temperature, see [49] for more
details. After that many authors studied this subject and many results were obtained, see
for instance [6, 14, 15, 16, 24, 30, 34, 42, 46, 49, 52, 56, 57] and references therein. More
specifically, many authors studied ways to estimate or determine exactly the eigenvalues
associated with the Stekloff problem and modifications of the latter, see [14, 56, 57]. By
following this way, we prove some upper and lower bounds for the Stekloff eigenvalues, see
Theorem 5.1, 5.2, 5.3, 5.4, 5.5. We point out that in the inequalities obtained in the Theorem
5.1, 5.2, 5.3 , 5.4 we characterize the equality cases. Moreover, in the 2-dimensional case, we
obtain a sharp result for weighted Stekloff eigenvalue problem, see Theorem 5.5.
In the setting of weighted Riemannian manifolds we study the following weighted Stekloff
eigenvalue problems:
#
∆f u “ 0
Bu
“ p1 u
Bν

in M,
on BM ;

#
∆2f u “ 0
u “ ∆f u ´ q1 Bu
“0
Bν

in M,
on BM ;

#
∆2f u “ 0
2
u “ BBνu2 ´ q1 Bu
“0
Bν

in M,
on BM,

where ν denotes the outward unit normal on BM . The first non-zero eigenvalue of the above
problems will be denoted by p1 and q1 , respectively. We will use the same letter for the first
non-zero eigenvalues for last two problems because whenever the weighted mean curvature
of BM is constant then the problems are equivalents.
Finally, in the Chapter 6, we obtain a weighted splitting theorem. Our inspiration to study
splitting theorems are the articles [11, 18], this latter using techniques from overdetermined
problems to obtain geometric restrictions over the space.
Our result says that, if M is a complete non-compact weighted Riemannian manifold with
Ricf ě 0 under suitable conditions, then M “ N ˆ R, where N is complete, totally geodesic,
and f -parabolic. Moreover, if u, g P C 8 pM q satisfy
∆f u ` gpuq “ 0,
12

where the function |∇u| satisfies
ż
|∇u|2 dVolf “ opR2 log Rq

as R Ñ `8,

BR

then
Volf pBRN q “ opR2 log Rq

as R Ñ `8,

and
żR

ˆ
1

2

|y ptq| dt “ o
´R

R2 log R
VolpBRN q

see Theorem 6.1.

13

˙
as R Ñ `8,

CHAPTER 2
PRELIMINARIES

In this first chapter we establish the concepts and definitions that we will use along this
thesis, as well as we also fix some notations.
Given any smooth positive function ϕ on a Riemanniann manifold pM, x , yq we can consider a new measure µ on M by formula dµ “ ϕdν, where ν is the Riemanniann measure.
The function ϕ is called the density function with respect to µ. For instance, the density
function of the Riemanniann measure ν is 1.
The triple pM, x , y, ϕq is called a weighted Riemanniann manifold, if pM, x , yq is a Riemanniann manifold and ν is a measure on M with density function ϕ. More generally, given
any f P C 8 pM q we can consider the density function ϕ :“ e´f , and thus dνf “ e´f dν, and
we will write pM, x , y, f q; some authors called pM, x , y, ϕq by Bakry-Émery manifold. That
concept is directly related to Ricci flow, mean curvature flow, theory of optimal transportation, see [17, 33] for a good overview of this subject. An important example of weighted
2
Riemannian manifold is the Euclidean space endowed with the Gaussian density e´π|x| , with
applications in probability and statistics.
In a weighted Riemannian manifolds there are natural generalizations for sectional, Ricci,
and scalar curvatures. With respect to the sectional curvature, William W. purpose in [53]
two new concepts of weighted sectional curvature as follow: given X, Y , unit orthogonal
vectors in Tp M him defined
secX
f pY q “ secf pX, Y q “ secpX, Y q ` Hessf pX, Xq
2
secX
f pY q “ secf pX, Y q “ secpX, Y q ` Hessf pX, Xq ` pdf pXqq

(2.1)

where secpX, Y q is the usual sectional curvature of the plane spanned by X, Y , and Hessf is
the Hessian of f . We point out that secf and secf are asymmetrical, that is, secf pX, Y q ‰
secf pY, Xq and secf pX, Y q ‰ secf pY, Xq. In [53], we can see that these notions of sectional
curvature come naturally from at least three places: the radial curvature equation, the second
variation of energy formula, and formula for Killing fields. Among the several interesting
results obtained by William W. we highlight, if pM n , x , y, f q is a simply connected weighted
14

Riemannian manifold of dimension n ą 2, and secf “ h or secf “ h for some function h,
then pM n , gq has constant sectional curvature.
The concept of weighted Ricci tensor on a weighted Riemannian manifold pM, x , y, f q was
defined by Bakry and Émery in [7] as follow
Ricf “ Ric ` Hessf,
where Ric denotes the Ricci tensor on the Riemannian manifold pM, x , yq. The tensor Ricf
is also known as Bakry-Émery Ricci tensor, and more generally the N -Bakry-Émery Ricci
tensor is
df b df
, for N ą 0.
(2.2)
RicN
f “ Ricf ´
N
Finally, the natural generalization para the scalar curvature S, was introduced by Perelman in [39] as follows
S8 “ S ` 2∆M f ´ |∇f |2
known as Perelman’s scalar curvature.
Now we explain a concept related with the extrinsic geometry of a submanifold. Consider
an oriented hypersurface Σ. Let ν be a unit normal vector field and let A be the second
fundamental form of Σ w.r.t N . In [23] M. Gromov introduced the weighted mean curvature
as
Hf “ H ` xν, ∇f y,
where ∇f denote the gradient of f in M , and H is the trace of the second fundamental form
A.
On a weighted Riemannian manifold pM, x , y, f q we can to define the f -Laplacian operator ∆f u “ ∆u ´ x∇f, ∇uy, that is a natural generalization of the Laplace-Beltrami operator
∆. In a complete weighted Riemannian manifold, we know that ∆f is essentially a selfadjoint operator with respect to the measure dνf “ e´f dν. The operator ∆f is also known
as diffusion operator and Drift Laplacian, and by simplicity, we will call it f -Laplacian. The
operator ∆f arises in probability theory, potential theory and harmonic analysis on complete
and non-compact weighted Riemannian manifolds. Moreover, the f -Laplacian appear in the
Ornstein-Uhlenbeck equation.

15

CHAPTER 3
THE HESSIAN OF THE DISTANCE FUNCTION

3.1

Introduction

Let pM, x , yq be a Riemannian manifold and p0 P M . The distance function in M with
reference point p0 is the function r : M Ñ R defined by rpxq “ dpp0 , xq. The classical
comparison result to the Hessian of the distance function state that if the radial sectional
curvature has a lower (resp. upper) bound of the form
secrad ě ´Gprpxqq

presp. secrad ď ´Gprpxqqq

then the Hessian of the distance function satisfies
ˆ
˙
h1 prq
h1 prq
px¨, ¨y ´ dr b drq
resp. Hess r ě
px¨, ¨y ´ dr b drq,
Hess r ď
hprq
hprq

(3.1)

for some appropriate function h, see Theorem 2.3 in [40]. Of course, by taking the tracing in
p3.1q we obtain comparison results to the laplacian of the distance function.
In the setting of weighted Riemannian manifolds, Impera and Rimoldi obtained in [28] a
result that generalizes the classical comparison theorem of the distance function, (Theorem
2.3 in [40]).
We remember below the notation of little o and big O
Definition 1 We say that f pxq “ Opgpxqq as x Ñ a if there exists a constant C such that
|f pxq| ď C|gpxq| in some punctured neighborhood of a, that is for x P pa ´ δ, a ` δqztau for
some value of δ.
f pxq
We say f pxq “ opgpxqq as x Ñ a if lim
“ 0. This implies that there exists a
xÑa gpxq
punctured neighborhood of a on which g does not vanish.

16

For more details and properties for notation of o and O see [20, page 391].
The following result play a important role in the proof of the Theorem 3.2:
Proposition 3.1 Let G be a continuous function on r0, `8q and let gi P ACp0, Ti q (Absolutely Continuous) be solutions of the Riccati differential inequalities
g12
g2
´ aG ď 0 g21 ` 2 ´ aG ě 0 a.e.in p0, Ti q
a
a
satisfying the asymptotic condition
a
gi ptq “ ` Op1q as t Ñ 0` ,
t
for some a ą 0. Then T1 ď T2 and g1 ptq ď g2 ptq in p0, T1 q.
g11 `

For a proof of the result above see [40, page 29].

3.2

Comparison Theorem to the Hessian of the Distance Function

Using the same technique that [28] with suitable adaptations, and the concept of sectional
curvature given by William W.
secX
f pY q “ secf pX, Y q “ secpX, Y q ` Hessf pX, Xq
we show that:
Theorem 3.2 Let pM m , x , y, f q be a complete m-dimensional weighted Riemannian manifold. Having fixed a reference point p0 P M , let rpxq “ distM px, p0 q and let Dp0 “ M zCutpp0 q
be the domain of the normal geodesic coordinates centered at p0 . Given a smooth even function
G on R, let h be the solution of the Cauchy problem
#
h2 ´ pc ` Gqh “ 0
cPR
(3.2)
1
hp0q “ 0, h p0q “ 1,
and let I “ r0, r0 q Ă r0, `8q be the maximal interval where h is positive. Suppose that the
radial curvature
secf ě ´Gprpxqq presp. ďq on Br0 pp0 q.
(3.3)
Furthermore, assume that
ηprq :“ x∇r, ∇f y ě ´θprq presp. ďq
`
for some θ P C 0 pR`
0 q and ηpsq “ op1q as s Ñ 0 . Let

Hessf rp¨ , ¨q :“ Hess rp¨ , ¨q ´

1
x∇r, ∇f yx¨ , ¨y,
m

if Hessf ď cx¨ , ¨y presp. ěq, then
Hessf rp¨ , ¨q ď

h1
1
tx¨ , ¨y ´ dr b drp¨ , ¨qu ` θprqx¨ , ¨y presp. ěq.
h
m
17

(3.4)

Proof. Firstly, we observed that Hessrp∇r, Xq “ 0 for all X P Tx M and x P Dp0 ztp0 u. In
fact, let γ be the geodesic parametrized by arch length issuing from p0 with γps0 q “ x, then
γ is an integral curve of ∇r so that γ 1 psq “ ∇rpγpsqq, this imply ∇∇r ∇rpxq “ ∇γ 1 ps0 q γ 1 “ 0
consequently
Hessrp∇r, Xq “ xX, ∇∇r ∇ry “ 0.
Since Hessf r is symmetric, Tx M has an orthonormal base consisting of eigenvectors of the
Hessf r. Denoting λmax pxq, and λmin pxq, respectively, the greatest and smallest eigenvalues of
the Hessf r in the orthonormal complement of ∇rpxq (the ∇rpxq is an eigenvector of the Hess r
associated to the eigenvalue 0), the theorem is equivalent to show that on pDp0 ztp0 uqXBr0 pp0 q
1

(i) if p3.3q and p3.4q hold with ě, then λmax ď hh prpxqq ` m1 θprq,
1

(ii) if p3.3q and p3.4q hold with ď, then λmin ě hh prpxqq ` m1 θprq.
We proof the item (i), and the item (ii) is analogous. Let x P Dp0 ztp0 u, and γ be the
minimizing geodesic joining p0 to x. We claim that, if p3.3q holds, then the function ψ “
η
q ˝ γ satisfies
pλmax ` m
#
ψ1 ` ψ2 ď c ` G
for a.e. s ą 0,
1
ψpsq “ s ` op1q, as s Ñ 0` .
Let φ :“ h1 {h, we have
φ1 ` φ2 “

hh2 ´ h1 2 h1 2
h2
`
“
“ c ` G,
h2
h2
h

and by L’Hospital rule
ˆ
˙
ˆ 1
˙
sh1 psq ´ hpsq
1
h
1
lim` φpsq ´
“ lim`
psq ´
“ lim`
sÑ0
sÑ0
sÑ0
s
h
s
shpsq
ˆ 1
˙
2
1
h psq ` sh psq ´ h psq
“ lim`
sÑ0
hpsq ` sh1 psq
˙
ˆ
spc ` Gqhpsq
“ lim`
sÑ0
hpsq ` sh1 psq
spc ` Gq
“ lim`
“ 0.,
sÑ0
2

(3.5)

that is,
φpsq ´

1
“ op1q.
s

Therefore, φ satisfies the following system
# 1
φ ` φ2 “ c ` G
1
φpsq “ ` op1q
s

on p0, r0 q and
as s Ñ 0`

follow by proposition 3.1 that
ψďφ“
18

h1
.
h

(3.6)

We will to show that p3.6q holds. Indeed,
and observe that, Therefore,
1
“ op1q.
s
Now, we will show that λmax have the required properties. To this end, given a smooth
real function u, denote by hessf u the p1, 1q symmetric tensor field defined by
φpsq ´

hessf upXq “ hessupXq ´

x∇f, ∇uyX
,
m

where
hessupXq “ ∇X ∇u
consequently, we have
B
F
x∇f, ∇uyX
xhessf upXq, Y y “ hessupXq ´
,Y
m
x∇f, ∇uy
“ xhessupXq, Y y ´
xX, Y y
m
x∇f, ∇uy
“ HessupX, Y q ´
xX, Y y
m
“ Hessf upX, Y q.

By definition of covariant derivative of tensors
∇X phessf uqpY q “ ∇X rhessf upY qs ´ hessf up∇X Y q
and
∇Y phessf uqpXq “ ∇Y rhessf upXqs ´ hessf up∇Y Xq.
Hence
∇X phessf uqpY q ´ ∇Y phessf uqpXq “ ∇X rhessf upY qs ´ hessf up∇X Y q´
´∇Y rhessf upXqs ` hessf up∇Y Xq

„
x∇f, ∇uyY
´ hessup∇X Y q`
“ ∇X hessupY q ´
m
„

x∇f, ∇uy∇X Y
x∇f, ∇uyX
`
´ ∇Y hessupXq ´
`
m
m
x∇f, ∇uy∇Y X
` hessup∇Y Xq ´
m
ˆ
˙
x∇f, ∇uy
x∇f, ∇uy
“ ∇X ∇Y ∇u ´
∇X Y ´ X
Y´
m
m
x∇f, ∇uy
´ ∇∇X Y ∇u `
rX, Y s ´ ∇Y ∇X ∇u`
m ˆ
˙
x∇f, ∇uy
x∇f, ∇uy
`
∇Y X ` Y
X ` ∇∇Y X ∇u
m
m

19

x∇f, ∇uy
rX, Y s`
∇X phessf uqpY q ´ ∇Y phessf uqpXq “ ∇X ∇Y ∇u ´ ∇Y ∇X ∇u ´ ∇rX,Y s ∇u ´
m
ˆ
˙
x∇f, ∇uy
x∇f, ∇uy
`
rX, Y s ´ X
Y`
m
m
ˆ
˙
x∇f, ∇uy
`Y
X,
m
from where
∇X phessf uqpY q ´ ∇Y phessf uqpXq “ RpX, Y q∇u´
ˆ
˙
ˆ
˙
x∇f, ∇uy
x∇f, ∇uy
´X
Y `Y
X.
m
m
Now, choose u “ rpxq, X “ ∇r and let γ be the minimizing geodesic joining p0 to
x P Dp0 ztp0 u. For every unit vector Y P Tx M such that Y K γ 1 ps0 q, where γps0 q “ x, define
a vector field Y K γ 1 , by parallel translation along γ. Since, as noted above, hessrp∇rq ” 0,
so
∇γ 1 ps0 q rhessf rpY qs “ ∇γ 1 ps0 q phessf rqpY q ` hessf rp∇γ 1 ps0 q Y q
“ ∇∇r phessf rqpY q “ ∇Y phessf rqp∇rq ` Rp∇r, Y q∇r´
˙
ˆ
˙
ˆ
x∇f, ∇ry
x∇f, ∇ry
Y `Y
∇r
´ ∇r
m
m
“ ∇Y rhessf rp∇rqs ´ hessf rp∇Y ∇rq ´ RpY, ∇rq∇r´
ˆ
˙
ˆ
˙
x∇f, ∇ry
x∇f, ∇ry
´ ∇r
Y `Y
∇r
m
m
„

x∇f, ∇ry∇r
“ ∇Y hessrp∇rq ´
´ hessrp∇Y ∇rq`
m
x∇f, ∇ry∇Y ∇r
`
´ RpY, ∇rq∇r´
˙
ˆ
˙
ˆ m
x∇f, ∇ry
x∇f, ∇ry
´ ∇r
Y `Y
∇r
m
m
ˆ
˙
x∇f, ∇ry
x∇f, ∇ry
“´
∇Y ∇r ´ Y
∇r´
m
m
x∇f, ∇ry∇Y ∇r
´ hessrp∇Y ∇rq `
´ RpY, ∇rq∇r´
m
ˆ
˙
ˆ
˙
x∇f, ∇ry
x∇f, ∇ry
´ ∇r
Y `Y
∇r
m
m
ˆ
˙
x∇f, ∇ry
Y
“ ´hessrp∇Y ∇rq ´ RpY, ∇rq∇r ´ ∇r
m
ˆ
˙
x∇f, ∇ry
“ ´hessrphessrpY qq ´ RpY, ∇rq∇r ´ ∇r
Y,
m
Since Y is parallel, we have
d
xhessf rpY q, Y y “ x∇γ 1 rhessf rpY qs, Y y,
ds
20

and consequently
d
pHessf rpγqpY, Y qq ` xhessf rpγqpY q, hessf rpγqpY qy “
ds B
´ hessrphessrpY qq ´ RpY, ∇rq∇r´
ˆ
˙
F
x∇f, ∇ry
´ ∇r
Y, Y `
m
F
B
x∇f, ∇ry
x∇f, ∇ry
Y, hessrpY q ´
Y
` hessrpY q ´
m
m
“ ´xhessrphessrpY qq, Y y ´ xRpY, ∇rq∇r, Y y´
ˆ
˙
x∇f, ∇ry
´ ∇r
` xhessrpY q, hessrpY qy´
m
2x∇f, ∇ry
x∇f, ∇ry2
´
xhessrpY q, Y y `
m
m2 ˙
ˆ
x∇f, ∇ry
“ ´xRpY, ∇rq∇r, Y y ´ ∇r
´
m
2x∇f, ∇ry
x∇f, ∇ry2
´
xhessrpY q, Y y `
m
m2
1
“ ´ secpY, γ q “ Hessf pY, Y q ´ secf pY, γ 1 q
ď Hessf pY, Y q ` Gprq
ď c ` Gprq
“

By other hand,
d
1 d
d
rHessf rpγqpY, Y qs “ rHess rpγqpY, Y qs ´
x∇ r, ∇f y ˝ γ
ds
ds
m ds
d
1 d
“ rHess rpγqpY, Y qs ´
η ˝ γ.
ds
m ds
Observe that
hessf rpγqpY q “ hess rpγqpY q ´

1
pη ˝ γqY
m

from where, we have
xhessf rpγqpY q, hessf rpγqpY qy “ xhess rpγqpY q, hess rpγqpY qy´
1
2pη ˝ γq
Hess rpγqpY, Y q ` 2 pη ˝ γq2 ,
´
m
m

21

(3.7)
(3.8)

and consequently,
d
d
1 d
rHessf rpγqpY, Y qs ` khessf rpY qk2 “ rHess rpγqpY, Y qs ´
η ˝ γ`
ds
ds
m ds
2pη ˝ γq
1
` xhess rpγqpY q, hess rpγqpY qy ´
Hess rpγqpY, Y q ` 2 pη ˝ γq2
m
m
1 d
2pη ˝ γq
ď Hessf pY, Y q ` Gprq ´
η˝γ´
Hess rpγqpY, Y q`
m ds
m
1
` 2 pη ˝ γq2
m
1 d
2pη ˝ γq
“ Hessf pY, Y q ` Gprq ´
η˝γ´
Hessf rpγqpY, Y q´
m ds
m
pη ˝ γq2
´
.
m2

(3.9)

Note that, for all unit vector field X K ∇r,
Hessf rpX, Xq ď λmax .
In fact, choosing a base tv1 , . . . , vn´1 u of tγ 1 uK formed by eigenvalues of hessf r, and writing
n´1
ÿ

X“

ai vi

i“1

we obtain
Hessf rpX, Xq “ xhessf rpXq, Xy
C
˜
¸
G
n´1
n´1
ÿ
ÿ
“ hessf r
ai v i ,
ai vi
i“1

C

n´1
ÿ

“

ai λi vi ,

i“1

G
ai vi

i“1

C
ď λmax

i“1

n´1
ÿ

n´1
ÿ
i“1

ai v i ,

n´1
ÿ

G
ai vi

“ λmax .

(3.10)

i“1

Then, if Y is chosen such that, in s0
Hessf rpγqpY, Y q “ λmax pγps0 qq,
that is, Y is eigenvector of hessf r in γps0 q, then the function
Hessf rpγqpY, Y q ´ λmax ˝ γ
attains its maximum at s “ s0 and, if at this point λmax is differentiable, then its derivative
vanishes:
ˇ
ˇ
d ˇˇ
d ˇˇ
Hessf rpγqpY, Y q ´ ˇ λmax ˝ γ “ 0.
ds ˇs0
ds s0
22

consequently, using p3.9q, we obtain in s0
d
1 d
pλmax ˝ γq ` pλmax ˝ γq2 ď Hessf pY, Y q ` Gprq ´
pη ˝ γq´
ds
m ds
pη ˝ γq2
2pη ˝ γq
Hessf rpγqpY, Y q ´
´
m
m2
d η˝γ
pη ˝ γq pη ˝ γq2
“ Hessf pY, Y q ` Gprq ´
´ 2pλmax ˝ γq
´
ds m
m
m2
η
Now, let ψ “ pλmax ` m q ˝ γ, from where
d
d pη ˝ γq
pη ˝ γq
pλmax ˝ γq `
` pλmax ˝ γq2 ` 2pλmax ˝ γq
`
ds
ds m
m
pη ˝ γq2
`
m2
ď Hessf pY, Y q ` Gprq
ď c ` Gprq

ψ1 ` ψ2 “

this is the desired inequality. The asymptotic behavior of ψ near s “ 0` follows from the
fact that
1
(3.11)
Hessr “ px¨ , ¨y ´ dr b drq ` op1q, r Ñ 0` ,
r
and from the assumptions about η. In fact, since Y is unit and Y K γ 1 “ ∇r, we have
η
η
ψ “ λmax ˝ γpsq `
˝ γpsq “ HessrpγpsqqpY, Y q `
˝ γpsq
m
m
1
1
“ pxY, Y y ´ x∇r, Y yx∇r, Y yq ` op1q ` op1q
s
m
1
“ ` op1q.
s
Therefore,
λmax ď

η
h1
1
h1
´
ď ` θ,
h
m
h
m

consequently, using p3.10q,
h1
1
Hessf rpX, Xq ď λmax xX, Xy ď xX, Xy ` θprqxX, Xy.
h
m
By other hand, if X “ X1 ` X2 where X1 {{∇r, and X2 K ∇r, we have
Hessf rpX, Xq “ Hessf rpX1 ` X2 , X1 ` X2 q
“ Hessf rpX1 , X1 q ` Hess
f rpX1 , X2 q ` Hess
f rpX2 , X1 q `Hessf rpX2 , X2 q
loooooooomoooooooon
loooooooomoooooooon
“0

“0

1

h
1
1
xX2 , X2 y ` θprqxX2 , X2 y ´ x∇r, ∇f yxX1 , X1 y
h
m
m
h1
1
1
ď txX, Xy ´ xX1 , X1 yu ` θprqxX2 , X2 y ` θprqxX1 , X1 y
h
m
m
h1
1
“ txX, Xy ´ dr b drpX, Xqu ` θprqxX, Xy
h
m

ď

23

and this conclude the proof.
Remark 3.3 Note that, if we assume secf ě ´G presp. ďq in the place of secf , we conclude
from 3.7 that the theorem holds.
Corollary 3.1 In the same assumptions of the theorem, we have
h1
∆f r ď pm ´ 1q prq ` θprq
h

presp. ěq.

Remark 3.4 The corollary 3.1 recover comparison results for weighted Riemannian manifolds with Ricf p∇r, ∇rq ě ´pm ´ 1qGprq and f satisfying 3.4 for some non-decreasing
function θ P C 0 r0, `8q, see [43, Theorem 3.1].
Corollary 3.2 Let Rm be the euclidean space m-dimensional with weighted f “ kxk2 {2.
Given a smooth even function G on R, let h be the solution of the Cauchy problem
#
h2 ´ p1 ` Gqh “ 0
hp0q “ 0, h1 p0q “ 1,
and let I “ r0, r0 q Ă r0, `8q be maximal interval where h is positive. Suppose that the radial
curvature satisfies
secf ě ´Gprpxqq presp. ďq on Br0 p0q.
(3.12)
Let
Hessf rp¨ , ¨q :“ Hess rp¨ , ¨q ´
then
Hessf rpxqp¨ , ¨q ď

1
x∇r, ∇f yx¨ , ¨y,
m

h1
1
tx¨ , ¨y ´ dr b drp¨ , ¨qu `
kxkx¨ , ¨y presp. ěq.
h
2m

Proof. We have that ηprpxqq “ x∇r, ∇f y “ kxk{2, we choose θprpxqq “ }x}{2, and is clear
that
lim` ηpsq “ 0 “ op1q.
sÑ0

Since secpV, U q “ 0, we have
secf pV, U q “ Hessf pV, V q “ xV, V y “ }V }2 ,
choosing Gpsq “ s2 , the result follow by the Theorem 3.2.
Theorem 3.5 Let pM m , x , y, f q be a complete weighted Riemannian manifold, and f super
harmonic. Assume that the radial Bakry-Émery-Ricci tensor of M satisfies
Ricf p∇r, ∇rq ě ´pm ´ 1qGprq
24

(3.13)

for some function G P C 0 pr0, `8qq, and that ηprq “ x∇r, ∇f y let such that ηpsq “ op1q as
s Ñ 0` . Let h P C 2 pr0, `8qq a solution to the problem
" 2
h ´ Gh ě 0,
(3.14)
hp0q “ 0, h1 p0q “ 1.
then the inequality
∆f rpxq ď pm ´ 1q

h1 prpxqq
hprpxqq

(3.15)

hold pointwise on M zpCutpp0 q Y tp0 uq.
Proof. Let r0, r0 q Ă r0, `8q be the maximal interval where h is positive. Let Dp0 “
M zCutpp0 q and fix x P Dp0 X rBr0 pp0 qztp0 us. Let γ : r0, `s Ñ M be the minimizing geodesic
joining p0 to x parametrized by arch length. Define
ϕpsq “ p∆f rqpγpsqq,

s P p0, `s.

We claim that ϕ satisfies
"
` op1q,
piq ϕpsq “ m´1
s
1
piiq ϕ1 ` m´1
ϕ2 ď pm ´ 1qG,

as s Ñ 0` ,
on p0, `s.

(3.16)

Indeed, as ηpsq “ op1q when s Ñ 0` we have, using p3.11q, that
∆f r “

m´1
` op1q,
r

as r Ñ 0` .

that proof the item (i) of p3.16q.
To the item (ii), of p3.16q, we obtain in p3.7q that
d
pHessf rpγqpY, Y qq`xhessf rpγqpY q, hessf rpγqpY qy “
ds
“ Hessf pY, Y q ´ Sectf pY, γ 1 q.

(3.17)

Taking the trace of p3.17q, we obtain
d
p∆f r ˝ γq ` kHessf rk2 pγq “ ∆f pγq ´ Ricf p∇r, ∇rqpγq.
ds
Taking an orthonormal basis t∇r, . . . , em u, and using Cauchy-Schwarz we obtain
kHessf rk2 ě

p∆f rq2
,
m´1

therefore, using that ∆f ď 0, we obtain
d
p∆r ˝ γq2
p∆r ˝ γq `
ď ´Ricf p∇r, ∇rqpγq.
ds
m´1
25

(3.18)

Since Ricf p∇r, ∇rq ě ´pm ´ 1qGprq we have
ϕ1 `

1
ϕ2 ď pm ´ 1qGprq.
m´1

1

Now, let ψ “ pm ´ 1q hh , so
ψ1 `

1
hh2 ´ h1 2 pm ´ 1q2 h1 2
ψ 2 “ pm ´ 1q
`
2
m´1
m ´˙1 h2
ˆ h2
12
hh ´ h
h1 2
“ pm ´ 1q
`
h2
h2
2
h
“ pm ´ 1q 2
h
ě pm ´ 1qG.

Since ψ “ pm ´ 1qφ, follow of p3.5q that
ψpsq “

m´1
` op1q.
s

Therefore, by the Proposition 3.1, we have
ϕďψ

em D0 X pBr0 pp0 qztp0 uq,

that is,
∆f rpxq ď pm ´ 1q

h1 prpxqq
hprpxqq

in D0 X pBr0 pp0 qztp0 uq.

N
Remark 3.6 Since RicN
f ď Ricf . Then, if Ricf ě ´pm ´ 1qGprq the result remains valid
for any N ą 0.

26

CHAPTER 4
ON THE FIRST STABILITY EIGENVALUE OF SURFACES

4.1

Introduction

In this chapter we presented upper bounds for the first eigenvalue of the weighted Jacobi
operator on surface with constant weighted mean curvature. In particular we generalize a
result obtained by Schoen and Yau (see [45, Theorem 5.1]) on stable minimal surfaces in
3-Riemannian manifolds with non-negative scalar curvature for the setting of weighted manifolds. We also show that, all closed λ-surface in Gaussian space are stable, and consequently
in R3 there is no closed stable self-shrinker.
Now, we introduce some objects related with the theory of surfaces in a weighted Riemannian manifold. Let Σ Ă M 3 be a two-sided surface of M 3 and consider N an unit normal
vector field globally defined on Σ. We will denote by A its second fundamental form and by
H the mean curvature of Σ, that is, the trace of A.
Following [53], we will use (2.1) slightly modified, as follow
ˆ
˙
1
pdf pXqq2
2m
secf pX, Y q “ secpX, Y q `
Hessf pX, Xq ´
,
2
2m

(4.1)

where X and Y are unit and orthogonal vectors fields tangents to M , and secpX, Y q is the
usual sectional curvature of the plane spanned by X and Y .
Taking N “ 2m in (2.2) we have
Ric2m
f “ Ricf ´

df b df
,
2m

(4.2)

where m ą 0.
We remember that, the natural generalization for the scalar curvature S of a weighted
Riemannian manifold M , is
S8 “ S ` 2∆M f ´ |∇f |2 ,
(4.3)

27

known as Perelman’s scalar curvature, see [39] for a good overview. We point out that S8 is
not the trace of Ric2m
for any m ą 0, and is not the trace of Ricf .
f
Throughout this chapter, dνf “ e´f dν will denote the weighted measure of the surface
Σ, where dν is the Riemannian measure of Σ, |Σ| and |Σ|f denote the area of Σ with respect
to the Riemannian measure and weighted measure of Σ, respectively. Furthermore, we will
denote by K the Gaussian curvature of Σ and by secΣ the sectional curvature of M restricted
to Σ.
It is a remarkable fact that, in the variational setting, surfaces with constant weighted
mean curvature are stationary points of the weighted area functional under variations that
preserves the weighted volume (see [8]). Moreover, the second variation of the weighted area
gives rise the weighted Jacobi operator on Σ, see [17], which is defined by
Jf u “ ∆f u ` p|A|2 ` Ricf pN, N qqu,

(4.4)

for any u P C 8 pΣq and |A|2 is the square Hilbert-Schmidt norm of A.
Now we will introduced the notion of stability with the goal of present some consequences.
Definition 2 Under the above notation. We say that a surface Σ is stable if the first eigenvalue λ1 of the weighted Jacobi operator is nonnegative. Otherwise, we say that Σ is unstable.

4.2

Estimates for the First Eigenvalue of the Weighted
Jacobi Operator

In this section we present the main results of this chapter. Our first result reads as follows
Theorem 4.1 Let pM 3 , x , y, f q be a weighted Riemannian manifold with S8 ě 6c, for some
c P R. Let Σ2 Ă M 3 be a closed surface with constant weighted mean curvature Hf . Denote
by λ1 the first eigenvalue of the weighted Jacobi operator. Then,
4πpg ´ 1q
1
.
λ1 ď ´ pHf2 ` 6cq ´
2
|Σ|
Moreover, equality holds if and only if Σ is totally geodesic, f is constant, S|Σ “ 6c and K
is constant.
Remark 4.2 In Riemannian case, f “ 0, the estimate can be improved. See Corollary 4.3
in the subsection 4.4.1.
The next result is a generalization of a result of Schoen and Yau on stable minimal surfaces
(see [45]) and this technique allow us to give an improvement of Theorem 2.1 in [19].
The result is the following:
Corollary 4.1 Let pM 3 , x , y, f q be a weighted Riemannian manifold with nonnegative Perelman’s scalar curvature. Let Σ be a closed stable surface with constant weighted mean curvature
Hf . Then Σ is conformally equivalent to the sphere S2 or Σ is a totally geodesic flat torus
T2 . Moreover, if S8 ą 0, then Σ is conformally equivalent to the sphere S2 .
28

The second result is the following:
Theorem 4.3 Let pM 3 , x , y, f q be a weighted Riemannian manifold with sec2m
f ě c, for some
c P R, and Hessf ď σ ¨ g for some real function σ on M . Let Σ2 Ă M 3 be a closed surface
with constant weighted mean curvature Hf . Denote by λ1 the first eigenvalue of the weighted
Jacobi operator. Then,
´ H2
¯
f
` 4c , with equality if and only if Σ is totally umbilical in M 3 , Ric2m
(i) λ1 ď ´ 12 1`m
f “
m
2c and df pN q “
Hf on Σ;
1`m
ş
ˆ
˙
Hf2
σ dνf
2 ş
Σ
(ii) λ1 ď ´
´ 4c ´
`
K dνf .
p1 ` 2mq
|Σ|f
|Σ|f Σ
2m
Moreover, if equality holds, then sec2m
“ c, Ric2m
“ 2c, df pN q “ 1`2m
Hf , and |A|
f
f
3
is a constant on Σ. Moreover, M has constant sectional curvature k and e´f is the
restriction of a coordinate function from the appropriate canonical embedding of a space
form Q3k in E4 , where E4 is R4 or L4 .

Our third result reads as follows:
Theorem 4.4 Let pM 3 , x , y, f q be a weighted Riemannian manifold with sec ě c, for some
df b df
pin the sense of quadratic formsq. Let Σ2 Ă M 3 be a closed
c P R, and Hessf ě
2m
surface with constant weighted mean curvature Hf . Denote by λ1 the first eigenvalue of the
weighted Jacobi operator. Then,
¯
1 ´ Hf2
(i) λ1 ď ´
` 4c , with equality if and only if Σ is totally umbilical in M 3 , RicpN, N q “
2 1`m
df pN q2
m
Hf on Σ and Hessf pN, N q “
;
2c, df pN q “
1`m
2m
Hf2
2 ş
´ 4c `
K dνf . Furthermore, equality holds if and only if K is
p1 ` 2mq
|Σ|f Σ
m
df pN q2
constant, secΣ “ c, df pN q “
Hf on Σ and Hessf pN, N q “
.
1`m
2m

(ii) λ1 ď ´

Remark 4.5 We believe that the hypotheses on the function f in theorems 4.3 and 4.4 are
natural, because we recovered the Riemannian case if the function is constant and also, for
m large enough, we captured the Gaussian space, which is very important in literature.
Now, we will give an application on the context of the mean curvature flow. For that, we
recall that a self-shrinker of the mean curvature flow is an oriented surface Σ Ă R3 such that
1
H “ ´ xx, N y,
2
29

where N is an unit normal vector field on Σ. The simplest examples of self-shrinkers in R3
2
are
of radius 2, and the cylinder S1 ˆ R1 , where the S1 has radius
? the plane R , the sphere
2
2. So, if we consider R3 endowed with the function f pxq “ |x|4 , then a self-shrinker is a
f -minimal surface in the Euclidean space. More generally, the triple pR3 , δij , |x|2 {4q is known
as Gaussian space and the surfaces with weighted mean curvature λ are know as λ-surfaces.
The next result is a consequence of the proof of the Theorem 4.4 and it reads as follows:
Corollary 4.2 All closed λ-surfaces in the Gaussian space are unstable. In particular, there
exist no closed stable self-shrinker surfaces in R3 .
This chapter is organized in this way: In section 4.3 we give a classification of weighted
Riemannian manifolds with constant weighted sectional curvature, we present also a way to
describe the first eigenvalue of the weighted Jacobi operator and, to conclude the section, we
rewrite the terms of the weighted Jacobi operator in an appropriate manner. In section 4.4
we present the proof of the results and the other consequences of them.

4.3

Preliminaries

An important result for us is the classification of weighted Riemannian manifolds with
constant weighted sectional curvature. The result below follows closely the one in [53], and
we include the proof here for the sake of completeness.
“ c,
Lemma 1 Let pM 3 , x , y, f q be a weighted Riemannian manifold. Assume that sec2m
f
then M has constant sectional curvature k, for some k P R. Moreover, c “ ´pm ´ 1qk and if
f is a non constant function, then u “ e´f {m is the restriction of a coordinate function from
the appropriate canonical embedding of a space form of curvature k, Q3k , in E4 , where E4 is
R4 or L4 .
Proof. Let X and Y be an unit and orthogonal vectors on M . Then, by equation (4.1), we
get
ˆ
˙
1
pdf pXqq2
c “ secpX, Y q `
Hessf pX, Xq ´
2
2m
and

1
c “ secpY, Xq `
2

ˆ

pdf pY qq2
Hessf pY, Y q ´
2m

So, there exists a smooth function w : M Ñ R such that
Hessf ´

df b df
“ w ¨ g.
2m

Then, letting tE1 , E2 , Xu be an orthonormal frame we have
2c “

2
ÿ

sec2m
f pX, Ei q “ RicpX, Xq ` 2w.

i“1

30

˙
.

Thus, by Schur’s Lemma, w is a constant function and so M has constant sectional
curvature, say k. Defining the function u “ e´f {m , we have that
Hessu “ ´

c´k
u ¨ g.
m

(4.5)

So, by Lemma 1.2 in [50],
g “ dt2 ` pu1 q2 g0 ,

(4.6)

where g0 is a local metric on a surface orthogonal to ∇u (a level set of u) and u1 denote the
derived of u in the direction of the gradient of u.
Computing the radial sectional curvature of the metric (4.6), we have pc`pm´1qkqu1 “ 0.
Since f is non constant, we have that c “ ´pm ´ 1qk. Moreover, as u satisfies equations
(4.5) and (4.6), u is the restriction of a coordinate function from the appropriate canonical
embedding of Q3k in E4 , where E4 is R4 or L4 .
Now we will describe the first stability eigenvalue in an appropriate manner. For this,
consider a first eigenfunction ρ P C 8 pΣq of the weighted Jacobi operator Jf , that is, Jf ρ “
´λ1 ρ; or equivalently,
∆f ρ “ ´pλ1 ` |A|2 ` Ricf pN, N qqρ.
(4.7)
Furthermore, λ1 is simple and it is characterized by
*
" ş
´ Σ uJf u dνf
8
ş
: u P C pΣq, u ‰ 0 .
λ1 “ inf
u2 dνf
Σ

(4.8)

We observe that the first eigenfunction of an elliptic second-order differential operator
has a sign. Therefore, without loss of generality, we can assume that ρ ą 0.
Thus,
∆f ln ρ “
“
“
“
“
“
“

∆ ln ρ ´ x∇f, ∇ln ρy
divΣ p∇ln ρq ´ x∇f, ρ´1 ∇ρy
divΣ pρ´1 ∇ρq ´ ρ´1 x∇f, ∇ρy
ρ´1 divΣ p∇ρq ` x∇ρ´1 , ∇ρy ´ ρ´1 x∇f, ∇ρy
ρ´1 p∆ρ ´ x∇f, ∇ρyq ´ ρ´2 |∇ρ|2
ρ´1 ∆f ρ ´ ρ´2 |∇ρ|2
´pλ1 ` |A|2 ` Ricf pN, N qq ´ ρ´2 |∇ρ|2 .

(4.9)

Integrating the equality above on Σ with respect to the weighted measure dνf and using the
divergence theorem we have that
ż
0 “ ´λ1 |Σ|f ´ p|A|2 ` Ricf pN, N qq dνf ´ α,
Σ

ş

where α :“ Σ ρ´2 |∇ρ|2 dνf ě 0 defines a simple invariant that is independent of the choice
of ρ, because λ1 is simple. So,
ż
1
λ1 “ ´
pα ` p|A|2 ` Ricf pN, N qq dνf q.
(4.10)
|Σ|f
Σ
31

Let tEi u be an orthonormal frame in T Σ and taij u the coefficients of A in the frame,
using the Gauss equation
K “ secΣ ´xApXq, Y y2 ` xApXq, XyxApY q, Y y,
we have that
1
K ´ secΣ “ a11 a22 ´ a212 “
2

˜

2
ÿ

pa11 ` a22 q2 ´

i,j“1

¸
a2ij

“

˘
1` 2
H ´ |A|2 ,
2

hence
|A|2 “ H 2 ` 2psecΣ ´Kq.

(4.11)

To complete this section, we recall the traceless of the second fundamental form of Σ,
that is, the tensor φ defined by φ “ A ´ H2 I, where I denote the identity endomorphism on
2
T Σ. We note that trpφq “ 0 and |φ|2 “ |A|2 ´ H2 ě 0, with equality if and only if Σ is totally
umbilical, where |φ|2 is the Hilbert-Schmidt norm.
In the literature, φ is know as the total umbilicity tensor of Σ. In terms of φ, the weighted
Jacobi operator is rewritten as
ˆ
˙
H2
2
Jf u “ ∆f u ` |φ| `
` Ricf pN, N q u.
(4.12)
2
We use exactly this expression in next section to obtain estimates of the first eigenvalue
of the weighted Jacobi operator.

4.4

Proof of the Theorems 4.1, 4.3 and 4.4

4.4.1

Proof of the Theorem 4.1

We start doing a straightforward calculus. Let te1 , e2 , e3 u be a adapted referential of Σ to
M . Lets rewrite the expression |A|2 ` Ricf pN, N q. We know that
S
“ secΣ `Ricpe3 q,
2
where S is the scalar curvature of M . By Gauss equation (4.11), we have
secΣ “ K ´

H 2 |A|2
`
.
2
2

32

Thus,
S
H 2 |A|2
´K `
`
` Hessf pe3 , e3 q
2
2
2
H 2 |A|2
1
1
S8 ´ ∆M f ` |∇f |2 ´ K `
`
` Hessf pe3 , e3 q
“
2
2
2
2
1
1
“
S8 ´ p∆Σ f ´ Hf3 ` Hessf pe3 , e3 qq ` p|∇f |2 ` f32 q
2
2
2
2
H
|A|
´K `
`
` Hessf pe3 , e3 q
2
2
1
1
1
1
S8 ´ K ´ ∆Σ f ` |∇f |2 ` Hf2 ` |A|2 .
(4.13)
“
2
2
2
2

|A|2 ` Ricf pN, N q “

Integrating with respect to Riemannian measure dν, using the divergence theorem and GaussBonnet theorem we obtain
ż
ż
1
2
|A| ` Ricf pN, N q dν “ 4πpg ´ 1q `
pS8 ` Hf2 ` |A|2 ` |∇f |2 q dν.
2
Σ
Σ
By the other hand, integrating (4.9) with respect to dν we obtain that
ż
ż
1
´ x ∇ρ, ∇f ydν “ ´λ1 |Σ| ´ pα ` p|A|2 ` Ricf pN, N qqdνq,
Σ ρ
Σ
and so,
ż
ż
|∇ρ|2 |∇f |2
´ p
`
qdν ď ´λ1 |Σ| ´ pα ` p|A|2 ` Ricf pN, N qqdνq.
2
2ρ
2
Σ
Σ
After a straightforward computation we have that
ż
1 α
1
λ1 ď ´ p ` 4πpg ´ 1q `
pS8 ` Hf2 ` |A|2 qdνq.
|Σ| 2
2 Σ
By our hypothesis,
4πpg ´ 1q
1
.
λ1 ď ´ pHf2 ` 6cq ´
2
|Σ|
Moreover, if equality holds then α “ 0 and thus ρ and f are constants, Σ is totally geodesic,
S|Σ “ 6c and K is constant. The reciprocal is immediate.
In Riemannian case, f “ 0, we can improve the estimate in theorem 4.1. The result is
the following:
Corollary 4.3 Let pM 3 , x , yq be a Riemannian manifold with S ě 6c, for some c P R. Let
Σ2 Ă M 3 be a closed surface with constant mean curvature H. Then,
3
4πpg ´ 1q
λ1 ď ´ pH 2 ` 4cq ´
.
4
|Σ|
Moreover, equality holds if and only if Σ is totally umbilical, S|Σ “ 6c and K is constant.
33

Proof. The equation (4.13) can be rewrite, with f “ 0, in the following way
1
3
1
|A|2 ` RicpN, N q “ S ´ K ` H 2 ` |φ|2 .
2
4
2
After a straightforward computation we have that
ż
1
1
3
λ1 ď ´ pα ` 4πpg ´ 1q `
pS ` H 2 ` |φ|2 qdνq,
|Σ|
2 Σ
2
and so

4πpg ´ 1q
3
λ1 ď ´ pH 2 ` 4cq ´
.
4
|Σ|

Moreover, if equality holds then α “ 0 and thus ρ is constant, Σ is totally umbilical, S|Σ “ 6c
and K is constant. The reciprocal is immediate.

4.4.2

Proof of the Theorem 4.3

Before to initiate the proof, we will recall the generalized sectional curvature
ˆ
˙
1
pdf pXqq2
2m
Sectf pX, Y q “ secpX, Y q `
Hessf pX, Xq ´
,
2
2m
where X, Y are unit and orthogonal vectors fields on M .
Moreover,
Ric2m
f pX, Xq “

2
ÿ

2m

Sectf pX, Yi q.

i“1

Since
2m
2m
Ric2m
f pN, N q ` 2 secΣ ě Ricf pN, N q ` 2secf |Σ ´ Hessf pX, Xq,

where X is a vector field on Σ. So, if Hessf ď σg, using (4.15) we rewrite the expression
p4.10q by

λ1

(4.14)
"
*
ż
ż
` 2m
˘
1
´
α ´ 2 K dνf `
Ricf pN, N q ` 2sec2m
ď ´
f ´ σ dνf .
1 ` 2m |Σ|f
Σ
Σ
Hf2

Now, we are able to prove our result.
Proof. The item (i) is a consequence of Theorem 4.4 (i). To second item, we using the
expression in (4.14) and our hypotheses.
2m
Now, if equality holds, then α “ 0, Ric2m
“ 2c and Sectf “ c. By equality in the
f
inequality p4.15q, we obtain
2m
df pN q “
Hf ,
1 ` 2m
34

and so

2m
1
Hf “
Hf .
1 ` 2m
1 ` 2m
Moreover, α “ 0 imply that ρ is constant and of the equation p4.7q we have that |A|2 is also
a constant.
To conclude, we using the Lemma 1 to obtain that M 3 has constant sectional curvature
and e´f has the property enunciate in case of the equality.
H “ Hf ´

In the next subsection we will provide the prove of Theorem 4.4 and some consequences.

4.4.3

Proof of the Theorem 4.4

Using (4.11) in (4.10) we obtain that
*
"
ż
ż
1
2
λ1 “ ´
α ´ 2 K dνf ` rH ` 2 secΣ `Ricf pN, N qsdνf .
|Σ|f
Σ
Σ
So, using the definition of weighted mean curvature we have
"
*
ż
ż
ż
1
2
λ1 “ ´
α ´ 2 K dνf ` pHf ´ xN, ∇f yq dνf ` r2 secΣ `Ricf pN, N qsdνf .
|Σ|f
Σ
Σ
Σ
Moreover, we know that for all a, b P R and k ą ´1, it holds that
b2
a2
´ ,
(4.15)
1`k
k
k
with equality if and only if b “ ´ 1`k
a. Applying that inequality with k “ 2m, using the
2m
definition of Ricf and a straightforward computation, we obtain that
"
*
ż
ż
` 2m
˘
Hf2
1
´
λ1 ď ´
α ´ 2 K dνf `
Ricf pN, N q ` 2 secΣ dνf .
1 ` 2m |Σ|f
Σ
Σ
pa ` bq2 ě

Using the hypotheses we obtain
Hf2
2
´ 4c `
λ1 ď ´
1 ` 2m
|Σ|f

ż
K dνf .

(4.16)

Σ

Proof. (i) Choosing the constant function u “ 1 to be the test function in p4.8q to
estimate λ1 , and using the expression in p4.12q, we obtain that
ş
„ż

ż
ż
´ Σ 1Jf 1 dνf
1
1
2
2
ş
λ1 ď
“´
|φ| dνf `
H dνf ` Ricf pN, N qdνf
|Σ|f Σ
2 Σ
1 dνf
Σ
Σ
„ż

ż
ż
1
1
pHf ´ xN, ∇f yq2 dνf ` Ricf pN, N qdνf
“´
|φ|2 dνf `
|Σ|f Σ
2 Σ
Σ
„ż
˙ ż

ż ˆ
2
2
Hf
1
1
xN, ∇f y
2
ď´
|φ| dνf `
´
dνf ` Ricf pN, N qdνf
|Σ|f Σ
2 Σ 1`m
m
Σ
ż
2
Hf
1
ď´
´ 2c ´
|φ|2 dνf
2p1 ` mq
|Σ|f Σ
ˆ
˙
Hf2
1
ď´
` 4c .
2 1`m
35

´ H2

¯
`
4c
, then all the inequalities above becomes equalities and conse1`m
m
quently Σ is totally umbilical, RicpN, N q “ 2c, df pN q “
Hf and Hessf pN, N q “
1`m
df pN q2
.
2m
m
On the other hand, if Σ is totally umbilical, RicpN, N q “ 2c, df pN q “
Hf and
1`m
2
df pN q
Hessf pN, N q “
, we have
2m
If λ1 “ ´ 12

f

H “ Hf ´ df pN q
m
“ Hf ´
Hf
1`m
1
“
Hf ,
1`m
and
1
pdf pN qq2
2m
m
“ 2c `
H 2.
2p1 ` mq2 f

Ricf pN, N q “ 2c `

Hence,
m
H2
` 2c `
H2
2
2p1 ` mq2 f
m
1
Hf2 ` 2c `
Hf2
“ ∆f `
2
2
2p1 ` mq
2p1 ` mq
1
“ ∆f `
H 2 ` 2c,
2p1 ` mq f

Jf “ ∆f `

and thus,
1
λ1 “ ´
2

ˆ

˙
Hf2
` 4c ,
1`m

as desired.
(ii) Using our hypotheses, we have by p4.16q that
Hf2
2
λ1 ď ´
´ 4c ´
1 ` 2m
|Σ|f

ż
K dνf .
Σ
2

pN q
. Firstly, we obtain of the
If equality holds, then α “ 0, secΣ “ c, Hessf pN, N q “ df2m
equation p4.15q that
2m
df pN q “
Hf .
1 ` 2m
1
and so H “
Hf . Moreover, α “ 0 implies ∇ρ “ 0 and thus using the equation p4.7q
1 ` 2m
we have that |A|2 is constant. Futhermore, by equation p4.11q, we have that K is constant.

36

On the other hand, if K is constant, secΣ “ c, Hessf pN, N q “
2m
Hf , we have that
1 ` 2m
2m
Ricf pN, N q “ 2c `
H 2,
p1 ` 2mq2 f
and so

df pN q2
and df pN q “
2m

Jf “ ∆f ` |A|2 ` Ricf pN, N q
“ ∆f ` H 2 ` 2pc ´ Kq ` 2c `
“ ∆f ` 4c `

2m
H2
p1 ` 2mq2 f

1
Hf2 ´ 2K,
1 ` 2m

and this implies that
λ1 “ ´4c ´

1
H 2 ` 2K.
1 ` 2m f

Now, using that K is constant,
Hf2
2
`
λ1 “ ´4c ´
1 ` 2m |Σ|f

ż
K dνf ,
Σ

as desired.
Now considering that the ambient is a 3-dimensional simply connected space form with
sectional curvature c, Q3c . If c is positive, we assume that all surfaces are contained in a
hemisphere. In that conditions we obtain the follows result:
Corollary 4.4 Let Σ Ă Q3c be a closed and orientable surface with constant weighted mean
curvature Hf , where f is one half of the square of the extrinsic distance?function. Assume
that Σ is contained in the geodesic ball center in the origin 0 and radius 2m. Then
ˆ
˙
Hf2
1
(i) λ1 ď ´
` 4c
2 1`m
(ii) λ1 ď ´

Hf2
2 ş
´ 4c `
K dνf .
p1 ` 2mq
|Σ|f Σ

The equalities holds if and only if Σ is the sphere center in the origin and radius
?
π
provided that 2m ď ? in case c ą 0.
c
Proof. We know that
pdr2 pN qq2
2m
2
Ricf rpN, N q “ 2c ` Hessr pN, N q ´
2m

and
Hessr2 pN, N q “ x∇N ∇r2 , N y
“ 2pdrpN qq2 ` 2rHessrpN, N q.
37

?
2m,

Now, using the expression of the Hessian of the distance function in a space form, we have
that
HessrpN, N q “ cotc prqr1 ´ pdrpN qq2 s,
where

$ ?
?
& ´c cothp ´csq if c ă 0,
1
if c “ 0,
cotc psq “
s ?
% ?
c cotp csq
if c ą 0.

So,
Hessr2 pN, N q “ 2pdrpN q2 q ` 2r cotc prqr1 ´ pdrpN qq2 s.
Now, using that the surface is contained in the ball center in the origin and radius
and pdrpN qq2 ď 1, we obtain that
2
2
Ric2m
f pN, N q “ 2c ` 2pdrpN qq ` 2r cotc prqp1 ´ pdrpN qq q ´

?
m

4r2 pdrpN qq2
2m

ě 2c.
Therefore, by Theorem 4.4, we conclude the inequalities enunciates. To conclude, if the
equalities holds, then drpN q “ 1 and r2 “ 2m.
Corollary 4.5 Let pM 3 , x , y, f q be a weighted Riemannian manifold with Ric2m
f ě 2c and
sec ě c.
(i) There is no closed stable surface with
Hf2
` 4c ą 0.
1`m
Hf2
` 4c ă 0, then
(ii) If Σ is a closed and stable surface such that
1 ` 2m
ż
Hf2
|Σ|f ě ´2p K dνf qp|
` 4c|q´1 .
1
`
2m
Σ
2

Proof. By definition, a surface is stable if and only if λ1 ě 0. Thus the item (i) follows from
the Theorem 4.4 (i). For the item (ii), we using the definition of stability and the Theorem
4.4 (ii). So,
ż
Hf2
2
´ 4c `
K dνf ,
0 ď λ1 ď ´
1 ` 2m
|Σ|f Σ
and thus

ˇ
ˇ
ż
ˇ
ˇ Hf2
ˇ
ˇ
` 4cˇ ě ´2 K dνf .
|Σ|f ˇ
1 ` 2m
Σ

Another consequence of the Theorem 4.4 is an improvement of the proposition 3.2 in [28]
for the case in that Σ is not necessarily f -minimal.
38

Corollary 4.6 Under the same assumptions of the Theorem 4.4.
(i) If c ą 0, then Σ cannot be stable;
(ii) If c “ 0, but Hf ‰ 0, then Σ cannot be stable;
(iii) If c “ 0 and Σ is stable, then Hf “ 0.

39

CHAPTER 5
STEKLOFF’S EIGENVALUES TO WEIGHTED RIEMANNIAN
MANIFOLDS

5.1

Introduction

The Classical Stekloff’s eigenvalue problem
#
∆u “ 0
Bu
“ σu
Bν

in Ω,
on BΩ,

was introduced by him in [49] for bounded domains Ω of the plane and afterward this was
studied by Payne in [42] for bounded domains in the plane with non-negative curvature.
In this chapter we study Stekloff’s eigenvalue problems in the weighted context. More
specifically, if pM, x , y, f q is a weighted Riemannian manifold with boundary BM , we study
the following weighted Stekloff’s eigenvalue problems
#
∆f u “ 0
Bu
“ pu
Bν

in M,
on BM ;

(5.1)

#
∆2f u “ 0
“0
u “ ∆f u ´ q Bu
Bν

in M,
on BM ;

(5.2)

#
∆2f u “ 0
2
u “ BBνu2 ´ q Bu
“0
Bν

in M,
on BM,

(5.3)

where ν denote the outward unit normal on BM . The first non-zero eigenvalues of the above
problems will be denoted by p1 and q1 , respectively. We will use the same letter for the first
40

non-zero eigenvalues of last two problems because whenever the weighted mean curvature of
BM is constant then the problems are equivalents, in the sense that u is solution of (5.2) if,
and only if, u it is solution of (5.3). Indeed, since in the boundary ∆BM u “ 0, we can write
∆M u “

B2u
Bu
,
`
nH
Bν 2
Bν

being u “ 0 on BM , and in this case ∇u “ Bu
ν, we have
Bν
Bu
xν, ∇f y
Bν
B2u
Bu
“ 2 ` pnH ´ xν, ∇f yq
Bν
Bν
Bu
B2u
“ 2 ` nHf .
Bν
Bν

∆f u “ ∆u ´ x∇u, ∇f y “ ∆u ´

(5.4)

Consequently,
Bu
B2u
Bu
“ 2 ´ pq1 ´ nHf q
Bν
Bν
Bν
therefore if Hf is constant, u is solution of (5.2) if, and only if, u it is solution of (5.3). Note
that, in this case, the difference between p1 and q1 is Hf .
∆f u ´ q1

In this chapter the N-Bakry-Émery Ricci tensor will be defined as
Rickf “ Ricf ´

df b df
,
k´n´1

(5.5)

where k ą n ` 1 or k “ n ` 1 and f a constant function. We will consider M n`1 a compact
oriented Riemannian manifold with boundary BM . Let i : BM ãÑ M be the standard
inclusion and ν the outward unit normal on BM . We will denote by II its second fundamental
form associate to ν, x∇X ν, Y y “ IIpX, Y q, and by H the mean curvature of BM , that is, the
trace of II over n.
In this chapter we will denote the weighted mean curvature, introduced by Gromov in
[23], of the inclusion i by
1
Hf “ H ´ xν, ∇f y.
n
This chapter is organized of the following way: in the section 5.2 we presented results
about upper bound and lower bound for the first non-zero Stekloff’s eigenvalue; in the section
5.3 we obtain the auxiliary results to proof the results of the previous section, in the section
5.4 we prove the four first results of the section 5.2, and finally in the section 5.5 we prove
the last theorem of the section 2.2.
Lastly, for the sake of simplicity, we will omit the weighted volume element in the integrals
in all text.

41

5.2

Weighted Stekloff ’s Eigenvalue Problems

In this section we presented our results. We point out that the Riemannian cases of following theorems was studied by Wang and Xia in [57, 58] and by Escobar in [14], respectively.
We start obtaining an upper bound for the first non-zero Stekloff’s eigenvalue of 5.1.
Theorem 5.1 Let M n`1 be a compact weighted Riemannian manifold with Rickf ě 0 and
boundary BM . Assume that the weighted mean curvature of BM satisfies Hf ě pk´1qc
, to
n
some positive constant c, and that second fundamental form II ě cI , in the quadratic form
sense. Denote by λ1 the first non-zero eigenvalue of the f -Laplacian acting on functions on
BM . Let p1 the first non-zero eigenvalue of the weighted Stekloff eigenvalue problem p5.1q.
Then,
?
a
λ1 a
(5.6)
p λ1 ` λ1 ´ pk ´ 1qc2 q
p1 ď
pk ´ 1qc
with equality occurs if and only if M is isometric to an n-dimensional euclidean ball of radius
1
, f is constant and k “ n ` 1.
c
The second result is the following:
Theorem 5.2 Let M n`1 be a compact connected weighted Riemannian manifold with Rickf ě
c,
0 and boundary BM . Assume that the weighted mean curvature of BM satisfies Hf ě k´1
k
to some positive constant c. Let q1 the first eigenvalue of the weighted Stekloff eigenvalue
problem p5.2q. Then
q1 ě nc.
Moreover, equality occurs if and only if M is isometric to a euclidean ball of radius 1c in
Rn`1 , f is constant and k “ n ` 1.
The next results are
Theorem 5.3 Let M n`1 be a compact connected weighted Riemannian manifold with boundary BM . Denote by A, V the weighted area of BM and the weighted volume of M , respectively.
Let q1 the first eigenvalue of the weighted Stekloff eigenvalue problem p5.2q. Then,
q1 ď

A
.
V

Moreover, if in addition that the Rickf of M is non-negative and that there is a point x0 P BM
, and q1 “ VA implies that M is isometric to an pn ` 1q-dimensional
such that Hf px0 q ě pk´1qA
knV
Euclidean ball, f is constant and k “ n ` 1.
and

42

Theorem 5.4 Let M n`1 be a compact connected weighted Riemannian manifold with Rickf ě
0 and boundary BM nonempty. Assume that Hf ě pk´1qc
, for some positive constant c. Let
n
q1 be the first eigenvalue of the problem p5.3q. Then
q1 ě c.
Moreover, equality occurs if and only if M is isometric to a ball of radius 1c in Rn`1 , f is
constant and k “ n ` 1.
Lastly, we announce a sharp estimate of the first non-zero Stekloff eigenvalue of surfaces
on suitable hypotheses.
Theorem 5.5 Let M 2 be a compact weighted Riemannian manifold with boundary. Assume
that M has non-negative Ricf , and that the geodesic curvature of BM, kg satisfies kg ´fν ě c ą
0. Let p1 the first non-zero eigenvalue of the Stekloff problem p5.1q. Assume that f is constant
on the boundary BM and its derivative in the direction normal exterior is nonnegative, then
p1 ě c. Moreover, the equality occur if and only if M is the Euclidean ball of radius c´1 and
f is constant.

5.3

Preliminaries

In this section we will present some results necessary to prove the theorems enunciated in
the previous section. We will present some proofs for the sake of completeness.
In [5] the authors proved the following useful inequality.
Proposition 5.6 Let u be a smooth function on M n`1 . then we have
|Hessu|2 ` Ricf p∇u, ∇uq ě

p∆f uq2
` Rickf p∇u, ∇uq,
k

for every k ą n ` 1 or k “ n ` 1 and f is a constant. Moreover, equality holds if and only if
∆u
Hessu “ n`1
x , y and x∇u, ∇f y “ ´ k´n´1
∆f u1 .
k
Proof. Let te1 , . . . , en`1 u be a orthonormal basis of Tp M , then by Cauchy-Schwarz inequality
we have that
p∆uq2 ď pn ` 1q|Hessu|2 .

(5.7)

1
1
Using that n`1
a2 ` k´n´1
b2 ě k1 pa ´ bq2 with equality if and only if

a“´
1

pn ` 1qb
,
k´n´1

This term only appear in the case of a non constant function.

43

(5.8)

we obtain
1
x∇f, ∇uy2
p∆uq2 ` Rickf p∇u, ∇uq `
n`1
k´n´1
1
ě p∆u ´ x∇f, ∇uyq2 ` Rickf p∇u, ∇uq
k
1
“ p∆f uq2 ` Rickf p∇u, ∇uq.
k

|Hessu|2 ` Ricf p∇u, ∇uq ě

(5.9)

If the equality holds, then since we use the Cauchy-Schwarz’s inequality in p5.7q we obtain
that Hessu “ λx , y, and by p5.8q
∆u “ ´

pn ` 1qx∇f, ∇uy
,
k´n´1

Consequently
∆f u “ ´

pn ` 1qx∇f, ∇uy
k
´ x∇f, ∇uy “ ´
x∇f, ∇uy.
k´n´1
k´n´1

The converse is immediate.
In [34] the authors showed that, for a smooth function u defined on an n-dimensional
compact weighted Riemannian manifold M with boundary BM , the following identity holds
, z “ u|BM and Ricf denote the generalized Ricci curvature of M :
if h “ Bu
Bν
ż
rp∆f uq2 ´|Hessu|2 ´ Ricf p∇u, ∇uqs “
(5.10)
M
ż
“
‰
nHf h2 ` 2h∆f z ` IIp∇z, ∇zq
“
BM

that is a generalization of the Reilly’s formula. Here, ∆ and ∇ represent the Laplacian and
the gradient on BM with respect to the induced metric on BM , respectively.
Using the Proposition 5.6 we have that
ż
k´1
rp∆f uq2 ´Rickf p∇u, ∇uqs ě
(5.11)
k
M
ż
ě
rnHf h2 ` 2h∆f z ` IIp∇z, ∇zqs.
BM

The next result is an estimate for the first non-zero eigenvalue of the f -Laplacian on
closed submanifolds.
Proposition 5.7 Let M n`1 be a compact weighted Riemannian manifold with nonempty
boundary BM and Rickf ě 0. If the second fundamental form of BM satisfies II ě cI, in the
quadratic form sense, and Hf ě k´1
c, then
n
λ1 pBM q ě pk ´ 1qc2 ,
where λ1 is the first non-zero eigenvalue of the f -Laplacian acting on functions on BM . The
equality holds if and only if M is isometric to an Euclidean ball of radius 1c , f is constant
and k “ n ` 1.
44

Proof. Let z be an eigenfunction corresponding to the first non-zero eigenvalue λ1 of the
f -Laplacian of BM , that is,
∆f z ` λ1 z “ 0.
(5.12)
Let u P C 8 pM q be the solution of the Dirichlet problem
#
∆f u “ 0
in M,
u“z
on BM.
It then follows from p5.11q and the non-negativity of Rickf of M that
ż
rnHf h2 ` 2h∆f z ` IIp∇z, ∇zqs.
0ě

(5.13)

BM

Since II ě cI, we have
IIp∇z, ∇zq ě c|∇z|2 ,
and noticing that

ż

ż

ż

2

BM

z2,

z∆z “ λ1

|∇z| “ ´
BM

BM

we obtain
ż
rnHf h2 ` 2h∆f z ` IIp∇z, ∇zqs

0ě
żBM

rpk ´ 1qch2 ´ 2λ1 zh ` cλ1 z 2 s
BM
ˆ
˙2
ˆ
˙ 
ż „
λ1 z
λ1
“
pk ´ 1qc h ´
` λ1 c ´
z2
pk ´ 1qc
pk ´ 1qc
BM
˙ż
ˆ
λ1
z2.
ě λ1 c ´
pk ´ 1qc
BM

ě

Consequently,
λ1 ě pk ´ 1qc2 ,
which proof the first part of theorem. The equality case follows by Proposition 5.6 and a
careful analysis in the equalities that occur. The converse is immediate.
Recall the following version of Hopf boundary point lemma, which will be important in
our proofs. See the proof in [26, Lemma 3.4].
Proposition 5.8 (Hopf boundary point lemma) Let pM n , x , yq be a complete Riemannian manifold and let Ω Ă M be a closed domain. If u : Ω Ñ R is a function with
u P C 2 pintpΩqq satisfying
∆u ` xX, ∇uy ě 0,
where X is a bounded vector field, x0 P BΩ is a point where
upxq ă upx0 q @x P Ω,
u is continuous at x0 , and Ω satisfies the interior sphere condition at x0 , then
Bu
px0 q ą 0
Bν
if this outward normal derivative exists.
45

5.4

Proof of Eigenvalue Estimates and Rigidity

In this section we will give the proof of the four first results announced in the introduction
and for this we will use all tools presented in the preliminaries.
Proof of Theorem 5.1. Let u be the solution of the following problem
#
∆f u “ 0
in M,
u|BM “ z,
where ˇz is a first eigenfunction corresponding to λ1 , that is, z satisfies ∆f z ` λ1 z “ 0. Set
ˇ , then we have from the Rayleigh inequality that (cf.[30])
h “ Bu
Bν BM
ş
h2
BM
ş
(5.14)
p1 ď
|∇u|2
M
and
ş

|∇u|2
z2
BM

ş
p1 ď M

(5.15)

Notice that p5.15q it is the variational principle, and p5.14q it is obtained as follows,
ş
ş
ş
|∇u|2
´ M u∆f u ` BM ux∇u, νy
M
ş
p1 ď ş
“
2
z
z2
BM
ş BM 2 ş
|∇u|
ux∇u, νy
BM
ş
ş
“ M
¨
z2
|∇u|2
BM
M
`ş
˘2
ux∇u,
νy
1
¨ BMş
“ş
2
z
|∇u|2
M
şBM 2 ş
z
x∇u, νy2
ş
ď şBM 2 ¨ BM
z
|∇u|2
BM
M
ş
h2
“ ş BM 2 ,
|∇u|
M
which gives

ş

h2
2
BM
ş
p1 ď
.
z2
BM
It then follows by substituting u into the equation p5.11q, we obtain
ż
k´1
0ě
rp∆f uq2 ´Rickf p∇u, ∇uqs ě
k
M
ż
ě
rnHf h2 ` 2h∆f z ` IIp∇z, ∇zqs
żBM
ě
rpk ´ 1qch2 ´ 2hλ1 z ` c|∇z|2 s.
BM

46

(5.16)

(5.17)

Note that, by Green’s formula,
ż
ż
2
|∇z| “
BM

ż

ż

x∇z, ∇zy “ ´

BM

BM

Putting this expression in (5.17) we have that
ż
ż
ż
2
0 ě pk ´ 1qc
h ´ 2λ1
hz ` cλ1
BM

BM
2

ě pk ´ 1qc

2

h ´ 2λ1

“

˙ 21 ˆż
z

«c

ż
h2 `
BM

2

˙ 12

ż
z2

` cλ1

BM

BM

pk ´ 1qc ´ λ1
c

z2

h

BM
2

BM

BM

ˆż

ż

z2.

z∆f z “ λ1

λ1
c

ˆż

c

λ1
c

h2

BM

˙ 12

a
´ cλ1

ˆż

BM

z2

˙ 21 ff2
,

BM

from where
a

and

λ1 ´ pk ´ 1qc2
?
c

ˆż
h2
BM

˙ 21
ě

ˆż
h2

˙ 12

ˆż
a
´ cλ1

BM

z2

BM

a
?
ˆż
˙ 12 a
ˆż
˙ 12
λ1 ´ λ1 ´ pk ´ 1qc2
2
2
?
ď cλ1
,
h
z
c
BM
BM

that is,
ˆż
2

˙ 12

h
BM

?
ˆż
˙ 21
c λ1
2
a
z
ď?
λ1 ´ λ1 ´ pk ´ 1qc2
BM
?
ˆż
˙ 12
a
λ1 a
2
“
z
.
p λ1 ` λ1 ´ pk ´ 1qc2 q
pk ´ 1qc
BM

Using p5.16q, we obtain
?
¯
a
λ1 ´a
2
p1 ď
λ1 ` λ1 ´ pk ´ 1qc .
pk ´ 1qc

Now, assume that
?
¯
a
λ1 ´a
2
p1 “
λ1 ` λ1 ´ pk ´ 1qc .
pk ´ 1qc
So, we also have that
ˆż
2

h
BM

˙ 12

?
˙ 12
¯ ˆż
a
λ1 ´a
2
2
“
λ1 ` λ1 ´ pk ´ 1qc
z
pk ´ 1qc
BM

47

˙ 12

and all inequalities above become equality. Thus h “ αz and
` 2ş
˘1
?
¯
a
α BM z 2 2
λ1 ´a
2
α “ `ş
λ1 ` λ1 ´ pk ´ 1qc ,
˘ 1 “ pk ´ 1qc
z2 2
BM

that is,

?
a
λ1 a
h“
p λ1 ` λ1 ´ pk ´ 1qc2 qz.
pk ´ 1qc
Furthermore we infer, by Proposition 5.6, that Hessu “ 0. Now, on the boundary BM , we
can write
∇u “ p∇uqJ ` p∇uqK
“ p∇uqJ ` x∇u, νyν,
where p∇uqJ is tangent to BM and p∇uqK is normal to BM . Then, take a local orthonormal
fields tei uni“1 tangent to BM . We obtain
0“

n
ÿ

Hessupei , ei q “

i“1
n
ÿ

“

n
ÿ

x∇ei ∇u, ei y

i“1

x∇ei rp∇uqJ ` x∇u, νyνs, ei y

i“1
n
ÿ

“

x∇ei p∇uqJ ` x∇u, νy∇ei ν ` ei px∇u, νyqν, ei y

i“1

“ ∆z `

n
ÿ

x∇u, νy IIpei , ei q

i“1

“ ∆z ` nHh
“ ∆f z ´ fν h ` nHh
“ ∆f z ` nHf h
“ ´λ1 z ` cpk ´ 1qh

?
a
λ1 a
p λ1 ` λ1 ´ pk ´ 1qc2 qz,
“ ´λ1 z ` cpk ´ 1q
pk ´ 1qc

from where
λ1 “ pk ´ 1qc2 .
Therefore, follow by Proposition 5.7, that M is isometric to an pn ` 1q-dimensional Euclidean ball of radius 1c , f is constant and so k “ n ` 1. The converse follows the ideas of the
Riemannian case.
Proof of Theorem 5.2. Let w be an eigenfunction corresponding to the first eigenvalue
q1 of problem p5.2q, that is
#
∆2f w “ 0
in M,
(5.18)
Bw
on BM.
w “ ∆f w ´ q1 Bν “ 0
48

Set η “ Bw
| ; then by divergence theorem we obtain
Bν BM
ż
ż
ż
2
p∆f wq “ ´ x∇p∆f wq, ∇wy `
∆f w x∇w, νy
M
M
BM
ż
ż
ż
“
w ∆f p∆f wq ´
w x∇p∆f wq, νy `
∆f w x∇w, νy
Mż
BM
BM
η2,
“ q1
BM

that is,

ş
q1 “

p∆f wq2
.
η2
BM

Mş

Substituting w in p5.11q, and noting that w|BM “ z, we have
ż
ż
ż
k´1
k
2
p∆f wq ě
Ricf p∇w, ∇wq `
nHf η 2
k
M
M
BM
ż
pk ´ 1qnc
η2,
ě
k
BM
from where q1 ě nc, as we desired.
Assume now that q1 “ nc, then the inequalities above become equalities and consequently
∆w
c. Furthermore, we have equality in the Proposition 5.6, thus Hessw “ n`1
x , y and
Hf “ k´1
k
k
∆f w “ n`1 ∆w.
Take an orthonormal frame te1 , . . . , en , en`1 u on M such that when restricted to BM
en`1 “ ν. Since w|BM “ 0 we have
ei pηq “ ei x∇w, νy
“ x∇ei ∇w, νy ` x∇w, ∇ei νy
“ Hesswpei , νq ` IIpp∇wqJ , ei q “ 0,
that is, η “ ρ “ constant, and so p∆f wq|BM “ q1 η “ ncρ is also a constant. Using the fact
that ∆f w is a f -harmonic function on M , we conclude by maximum principle that ∆f w is
k
∆w, then w satisfies
constant on M . Since ∆f w “ n`1
#
∆ w
Hessw “ kf x , y in M,
w|BM “ 0.
Thus, by Lema 3 in [48], we conclude that M is isometric to a ball in Rn`1 of radius c´1 .
∆ w
Now, using the hessian of w is possible see that w “ λ2 r2 ` C, where λ “ kf and r is the
distance function from its minimal point, see [48] for more details for this technique.
Lastly, we will show that f is constant. In fact, if k ą n ` 1, then x∇f, ∇wy is constant
and so f “ ´pk ´ n ´ 1q ln r ` C. It is a contradiction, since f is a smooth function.
Proof of Theorem 5.3. Now, let w be the solution of the following Laplace equation
#
∆f w “ 1 in M,
(5.19)
w|BM “ 0.
49

Follows from Rayleigh characterization of q1 that
ş
p∆f wq2
V
q1 ď Mş
“ş
,
(5.20)
2
η
η2
BM
BM
ˇ
ˇ . Integrating ∆f w “ 1 on M and using the divergence theorem, it gives
where η “ Bw
Bν BM
ż
V “
η.
BM

Hence we infer from Schwarz inequality that
ż
η2.

2

V ďA

(5.21)

BM

Consequently,
q1 ď ş

V

BM

η2

ď

V
A
“ .
2
V {A
V

Assume now that Rickf ě 0, Hf px0 q ě pk´1qA
for some x0 P BM and q1 “ VA . In this case
knV
p5.21q become a equality and so η “ VA is a constant. Consider the function φ on M given by
1
w
φ “ |∇w|2 ´ .
2
k
Using the Bochner formula p5.32q, ∆f w “ 1, the Proposition 5.6 and that Rickf ě 0, we have
that
1
1
∆f φ “ |Hessw|2 ` x∇w, ∇p∆f wqy ` Ricf p∇w, ∇wq ´
(5.22)
2
n`1
1
1
ě p∆f wq2 ´ “ 0.
k
k
` ˘2
Thus φ is f -subharmonic. Observe that φ “ 21 VA on the boundary. In fact, if we write
∇w “ p∇wqJ ` p∇wqK , where p∇wqJ is tangent to BM and p∇wqK is normal to BM , and
since w|BM “ 0, it follows that ∇w “ p∇wqK “ Cν on BM . On the other hand,
1 “ ∆f w “ q1 x∇w, νy “
Therefore φ “ 12

` V ˘2
A

A
V
V
C implies C “
and |∇w| “ .
V
A
A

on the boundary, and so we conclude by Proposition 5.8 that either
1
φ“
2

ˆ ˙2
V
A

in M

(5.23)

Bφ
pyq ą 0, @ y P BM.
Bν

(5.24)

or

50

From w|BM “ 0, we have
V
1 “ p∆f wq|BM “ nHη ` Hesswpν, νq ´ x∇f, νy
ˆ
˙A
nV
x∇f, νy
V
“
Hf `
` Hesswpν, νq ´ x∇f, νy
A
n
A
nV
Hf ` Hesswpν, νq.
“
A
Hence it holds on BM that
Bφ
V
V
“ Hesswpν, νq ´
Bν
Aˆ
˙k A
nV
V
V
1´
Hf ´
“
A
A
kA
ˆ
˙
V k´1
V
´ Hf
“n
,
A
kn
A
which shows that p5.24q is not true since Hf px0 q ě pk´1qA
. Therefore φ is constant on M .
knV
Since the f -Laplacian of φ vanishes, we infer that equality must hold in p5.22q and that give us
∆w
k
∆w and Hessw “ n`1
x , y.
equality in the Proposition 5.6, and consequently 1 “ ∆f w “ n`1
The remainder of the proof follows a similar arguments as in proof of Theorem 5.2.
Proof of Theorem 5.4. Let w be an eigenfunction corresponding to the first eigenvalue q1
of the problem p5.3q:
#
∆2f u “ 0
in M,
(5.25)
Bu
B2 u
on BM.
u “ Bν 2 ´ q Bν “ 0
Observe that w is not a constant. Otherwise, we would conclude from w|BM “ 0 that w ” 0.
| ; then η ‰ 0. In fact, if η “ 0 then
Set η “ Bw
Bν BM
B2w
“0
w|BM “ p∇wq|BM “
Bν 2
this implies, by (5.4), that p∆f wq|BM “ 0 and so ∆f w “ 0 on M by the maximum principal,
which in turn implies that w “ 0. This is a contradiction.
Since w|BM “ 0, we have by the divergence theorem that
ż
ż
x∇w, ∇p∆f wqy “ ´
w∆2f w “ 0,
M

hence

(5.26)

M

ż

ż
ż
ż
Bw
2
∆f w
“
x∇p∆f wq, ∇wy `
p∆f wq “
p∆f wq2 .
Bν
BM
M
M
M
Bw
Since w|BM “ 0, we have ∇w “ Bν ν and
B2w
Bw
` nH
´ x∇f, ∇wy
2
Bν
Bν
Bw
Bw
Bw
Bw
“ q1
` nHf
` x∇f, νy
´ x∇f, νy
Bν
Bν
Bν
Bν
Bw
Bw
“ q1
` nHf
Bν
Bν

p∆f wq|BM “

51

(5.27)

(5.28)

using p5.27q and p5.28q we obtain that
ş
q1 “

M

ş
p∆f wq2 ´ n BM Hf η 2
ş
.
η2
BM

On the other hand, substituting w into p5.11q, we obtain
ż
ż
ż
k´1
k
2
p∆f wq “
Ricf p∇w, ∇wq `
nHf η 2
k
M
BM
żM
ě
nHf η 2 ,

(5.29)

BM

that is,

ż

ż

n
p∆f wq ´
nHf η ě
k´1
M
BM
2

ż

ż
2

2

η2.

Hf η ě c
BM

BM

By expression for q1 and estimate above, we obtain the desired estimate
q1 ě c.

(5.30)

Assume now that q1 “ c. So all inequalities in p5.29q become equalities. Thus, by Proposition
5.6, we have that
Hessw “

∆w
x,y
n`1

and

∆f w “ ´

k
x∇f, ∇wy.
k´n´1

(5.31)

Choice an orthonormal frame te1 , . . . , en u on M so that restricted to BM, en “ ν. On the
other side, to i “ 1, . . . , n ´ 1, using that w|BM “ 0, we obtain
0 “ Hesswpei , en q “ ei en pwq ´ ∇ei en pwq
“ ei pηq ´ x∇ei en , en yη “ ei pηq,
follow that η “ b0 “ const. Since p5.30q takes equality and η is constant, we conclude
c, which implies from p5.28q that p∆f wq|BM “ kcb0 , therefore, by maximum
that Hf “ k´1
n
principle ∆f w is constant on M which implies from p5.31q that ∆w is constant on M . The
remainder of the proof follows a similar arguments as in proof of Theorem 5.2.

5.5

Sharp Estimate of the Stekloff ’s Eigenvalue for Surfaces

Recall the Bochner type formula for weighted Riemannian manifold, which says: Any
smooth function u on M holds that
1
∆f |∇u|2 “ |Hessu|2 ` x∇u, ∇p∆f uqy ` Ricf p∇u, ∇uq.
2

(5.32)

An immediate consequence of the Bochner type formula is the result below, however we
believe that this is not a sharp estimate.
52

Theorem 5.9 Let M n`1 , n ě 2 be a compact weighted Riemannian manifold with boundary
BM . Assume that Ricf ě 0, Hf ě 0 and that the second fundamental form satisfies II ě cI
on BM, c ą 0. Then
c
p1 ą .
2
, and z “ u|BM where u is solution of problem p5.1q. We have p1 z “ p1 u “
Proof. Set h “ Bu
Bν
h, thus p1 ∇z “ ∇h. By p5.10q, we have
ż
ż
2
0ą´
|Hessu| ě
rp∆f uq2 ´ |Hessu|2 ´ Ricf p∇u, ∇uqs
M
żM
“
‰
“
nHf h2 ` 2h∆f z ` IIp∇z, ∇zq
BMż
ż
|∇z|2
ě ´2
x∇h, ∇zy ` c
BM
ż
ż BM
2
ě ´2p1
|∇z| ` c
|∇z|2
BM

Note that

BM

ż
|∇z|2 ą 0.
BM

Otherwise z is constant on the Boundary and hence f is constant on M which is a contradiction. Thus p1 ą 2c .
Below we present the proof of the sharp estimate of the non-zero first Stekloff eigenvalue
on surfaces. The technique was introduced by Escobar in [15], and just allows us to attack
this problem in context of surfaces.
Proof of Theorem 5.5. Let φ be a non-constant eigenfunction for the Stekloff problem
p5.1q. Consider the function v “ 12 |∇φ|2 , then by p5.32q
∆f v “ |Hessφ|2 ` x∇φ, ∇p∆f φqy ` Ricf p∇φ, ∇φq.
Since φ is a f -harmonic function and Ricf ě 0 we find that
∆f v “ |Hessφ|2 ` Ricf p∇φ, ∇φq ě 0.

(5.33)

Therefore the maximum of v is achieved at some point P P BM . The Proposition 5.8 implies
that pBv{BηqpP q ą 0 or v is identically constant.
Let’s assume pBv{BηqpP q ą 0 and let pt, xq be Fermi coordinates around the point P , that is,
x represents a point on the curve BM and t represents the distance to the boundary point x.
The metric has the form
ds2 “ dt2 ` h2 pt, xqdx2 ,
(5.34)
where hpP q “ 1, pBh{BxqpP q “ 0. Thus
ˆ ˙2
ˆ ˙2
Bφ
Bφ
2
´2
|∇φ| “
`h
,
Bt
Bx
53

and
Bv
Bφ B 2 φ
Bφ B 2 φ
Bh
“
` h´2
´ h´3
2
Bx
Bt BxBt
Bx Bx
Bx

ˆ

Bφ
Bx

˙2
.

Evaluating at the point P we obtain
Bv
Bφ B 2 φ
Bφ B 2 φ
pP q “
`
“ 0.
Bx
Bt BxBt Bx Bx2

(5.35)

The f -Laplacian with respect to the metric given by p5.34q in Fermi coordinates pt, xq is
ˆ
˙
Bf B
Bf B
B2
´1 B
´1 B
´1 Bh B
`h
h
´
´ h´2
.
∆f “ 2 ` h
Bt
Bt Bt
Bx
Bx
Bt Bt
Bx Bx
The geodesic curvature of BM can be calculated in terms of the function f and its first
derivative as follows:
F
B
F
B
B B
B B
“ ´ ∇B{Bt ,
kg “ ´ ∇B{Bx ,
Bt Bx
Bx Bx
B
F
1B
B B
1B 2
“´
,
“´
ph q “ ´hh1 .
(5.36)
2 Bt Bx Bx
2 Bt
Hence at P we find that
B2φ
Bφ B 2 φ Bf Bφ Bf Bφ
0 “ ∆f φ “ 2 ´ kg
`
´
´
.
Bt
Bt
Bx
Bt Bt
Bx Bx

(5.37)

Using the equality p5.36q we get that
Bφ B 2 φ Bφ B 2 φ
Bv
pP q “
`
` kg
Bt
Bt Bt2
Bx BtBx

ˆ

Bφ
Bx

˙2
.

(5.38)

Multiplying the equation p5.37q by ´ Bφ
and adding with the equation p5.38q we obtain
Bt
Bv
Bφ B 2 φ Bφ B 2 φ
Bf
pP q “ kg |∇φ|2 ´
`
`
Bt
Bt Bx2 Bx BtBx
Bt

ˆ

Bφ
Bt

˙2
`

Bf Bφ Bφ
.
Bx Bx Bt

(5.39)

If Bφ
pP q ‰ 0, the equation p5.35q and the boundary condition yields
Bx
B2φ
Bφ
pP q “ p1 pP q.
2
Bx
Bt

(5.40)

Therefore the equation p5.39q can be re-written using the boundary condition as
ˆ ˙2
ˆ ˙2
Bφ
Bφ B 2 φ
Bf Bφ
Bf Bφ Bφ
Bv
2
pP q “ pkg ´ p1 q|∇φ| ` p1
`
`
`
.
Bt
Bx
Bx BtBx
Bt Bt
Bx Bx Bt

(5.41)

Notice that by p5.35q we obtain, using p5.40q,
Bφ B 2 φ
Bφ B 2 φ
Bφ
0“
`
“
2
Bt BxBt Bx Bx
Bt
54

ˆ

B2φ
Bφ
` p1
BxBt
Bx

˙
,

(5.42)

that is,
p1

Bφ
B2φ
“´
.
Bx
BxBt

Thus p5.41q becomes
Bv
Bf
pP q “ pkg ´ p1 q|∇φ|2 `
Bt
Bt

ˆ

Bφ
Bt

˙2
`

Bf Bφ Bφ
Bx Bx Bt

and we write
Bφ
Bv
pP q “ pkg ´ p1 q|∇φ|2 `
x∇φ, ∇f y.
Bt
Bt
Since f |BM is constant, so Bf
pP q “ 0, and using that Bf
ď0
Bx
Bt
ˆ ˙2
Bv
Bf Bφ
2
pP q “ pkg ´ p1 q|∇φ| `
Bt
Bt Bt
˙
ˆ
Bf
´ p1 |∇φ|2 ,
ě kg `
Bt
hence
pkg `

Bf
´ p1 q|∇φ|2 ă 0,
Bt

(5.43)

and p1 ą kg ` Bf
“ kg ´ fν ě c.
Bt
Now we assume that Bφ
pP q “ 0. A straighforward calculation yields
Bx
B2v
pP q “
Bx2

ˆ

B2φ
BxBt

˙2

Bφ B 3 φ
`
`
Bt Bx2 Bt

ˆ 2 ˙2
B φ
.
Bx2

Using the boundary condition we get that
2
B2v
2 B φ
pP
q
“
p
φ
`
1
Bx2
Bx2

ˆ 2 ˙2
B φ
ď 0.
Bx2

(5.44)

Since Bφ
pP q “ 0, the equation p5.39q implies that
Bx
ˆ ˙2
ˆ ˙2 ˆ
˙
Bv
Bφ
B 2 φ Bf Bφ
Bf
B2φ
pP q “ kg
` p1 φ 2 `
“ kg `
p21 φ2 ` p1 φ 2 .
Bt
Bt
Bx
Bt Bt
Bt
Bx
Thus

ˆ

Bf
kg `
Bt

˙
p31 φ2 ` p21 φ

B2φ
ă 0.
Bx2

Adding inequality p5.44q with p5.45q we obtain
ˆ
ˆ 2
˙2
˙
Bf
B φ
2
3
` p1 φ ` p1 kg `
´ p1 φ2 ă 0.
2
Bx
Bt
Hence
p1 ą kg `

Bf
“ kg ´ fν ě c.
Bt
55

(5.45)

Let’s assume that v is the constant function. Observe that v ı 0 because φ is nonconstant. Since v is f -harmonic, inequality p5.33q implies that
Hessφ “ 0

and

Ricf p∇φ, ∇φq “ 0,

on M.

Now, using that ∆f φ “ 0, we obtain that x∇φ, ∇f y “ 0 and hereby Hessf p∇φ, ∇φq “ 0.
Thus, the Gaussian curvature K of M vanishes. Moreover, using the structure of surfaces,
∇f “ λ Jp∇φq,

(5.46)

where J is the anti-clockwise rotation of π{2 in the tangent plane.
Let te1 , e2 u be a local orthonormal frame field such that e1 is tangent to BM and e2 “ η.
So,
0 “ Hessφpe1 , e2 q “ e1 e2 pφq ´ ∇e1 e2 pφq
“ e1 pp1 φq ´ x∇e1 e2 , e1 yφ1
“ pp1 ´ kg qφ1 .
Observe that if φ1 “ 0 on BM , then φ “constant on BM and hence φ is a constant function
on M which is a contradiction. Thus p1 “ kg except maybe when in the points where φ1 “ 0.
Since Hessφpe1 , e1 q “ 0 we have
0 “ Hessφpe1 , e1 q “ e1 e1 pφq ´ ∇e1 e1 pφq
“ e1 pe1 φq ´ x∇e1 e1 , e2 ye2 pφq
“ e1 pe1 φq ` kg p1 φ.
Hence φ satisfies on the boundary a second order differential equation
d2 φ
` kg p1 φ “ 0
dx2
φp0q “ φp`q

(5.47)

where ` represents the length of BM . The function φ does not vanishes identically, thus
φ1 “ 0 except for a finite number of points. Therefore p1 “ kg except for a finite number of
points and using the continuity of kg , we conclude that p1 “ kg everywhere. Therefore,
p1 “ kg ´ fν ` fν ě c,
and the equality between p1 and c occurs if kg “ k0 and fν “ 0. Using K “ 0 and kg is a
positive constant, we conclude that M is an Euclidean ball.
Furthermore, by the identity p5.46q, and using that Hessϕ “ 0, we obtain
∇X ∇f “ XpλqJp∇ϕq ` λJp∇X ∇ϕq
“ XpλqJp∇ϕq,
and note that
|∇f |2 “ λ2 |∇ϕ|2 “ 2λ2 v 2 ñ λ2 “
56

|∇f |2
.
2v 2

By other hand,
xJp∇X ∇ϕq, ∇ϕqy “ x∇X Jp∇ϕq, ∇ϕqy “ ´xJp∇ϕq, ∇X ∇ϕy.
Let te1 , e2 u be a orthonormal basis of the Tp M , then
∆f “

2
ÿ
i“1
2
ÿ

“

x∇ei ∇f, ei y
x∇λ, ei yxJp∇ϕq, ei y

i“1

“ x∇λ, Jp∇ϕqy,
and, using the symmetry of the Hessf , we obtain
x∇λ, ∇ϕy|Jp∇ϕq|2 “ x∇Jp∇ϕq ∇f, ∇ϕy “ x∇λ, ∇ϕyxJp∇ϕq, ∇ϕy “ 0.
Therefore, ∇λ “ ξJp∇ϕq and ∇X ∇f “ ξxJp∇ϕq, XyJp∇ϕq. Consequently,
Hessf “ ξJp∇ϕq b Jp∇ϕq
and
∆f “ ξ|Jp∇ϕq|2 “ 2ξv ñ ξ “
from where

∆f
,
2v

∆f
pJp∇φq b Jp∇φqq .
2v
It easy to see, using that M is an Euclidian ball, that φ “ xi , that is, φ is a coordinate
function. Thus, using the expression of φ, f satisfies Hessf “ 0 and as f is constant on the
boundary, we have f constant.
Hessf “

57

CHAPTER 6
A WEIGHTED SPLITTING THEOREM

6.1

Introduction

Given g P C 8 pM q we consider the closed Dirichlet problem
∆f u ` gpuq “ 0.

(6.1)

A solution of that problem is a critical point of an energy functional, which we will denote
by Ef . We say that a solution u is stable if the second variation of Ef is non-negative on
Wc1,2 pM q, where
Wc1,2 pM q “ tu P L2 pM q;

Bu
P L2 pM q, i “ 1, 2, . . . , m “ dimpM qu
Bxi

with compact support in M , see [13] for a good overview about Sobolev’s spaces.
We say that a weighted Riemannian manifold is f -parabolic if there exists no non-constant
and bounded below function which is f -superharmonic.
In this chapter our aim is to prove the following weighted splitting theorem. It is read as
follow:
Theorem 6.1 Let M be a complete and non-compact weighted Riemannian manifold without
boundary and satisfying Ricf ě 0. Assume that u P C 8 pM q is a non-constant and stable
solution of p6.1q.
If either
(i) M is f ´parabolic and ∇u P L8 pM q, or
(ii) the function |∇u| satisfies
ż
|∇u|2 dνf “ opR2 log Rq
BR

58

as R Ñ `8.

(6.2)

Then, M “ N ˆR with the product metric gM “ gN `dt2 , for some complete, totally geodesic,
2
1
f -parabolic hypersurface N . In particular, RicN
f ě 0 if m ě 3, and M “ R or S ˆ R, with
their flat metric, if m “ 2. Moreover, u depends only on t, has no critical points, and writing
u “ yptq it holds ´y 2 ` ky 1 “ gpyq where k is a constant.
Moreover, if (ii) is met,
Volf pBRN q “ opR2 log Rq
ˆ 2
˙
żR
R log R
1
2
|y ptq| dt “ o
Volf pBRN q
´R

as R Ñ `8.

(6.3)

as R Ñ `8.

(6.4)

This chapter is organized as follow; in section 2 we recall all concepts and equivalences
that we use in the chapter; in section 3 we present some technical propositions that will
auxiliary in the proofs of the principal results; in section 4 we dedicated it to proof of the
Theorem 6.1.

6.2

Preliminaries

Throughout the chapter M will denote a connect weighted Riemannian manifold of dimension m ě 2, without boundary. We briefly fix some notation. Having fixed an origin p0 ,
we set rpxq “ distpx, po q, and we write BR for geodesic ball centered at po . If we need to emphasize the set under consideration, we will add a superscript symbol, so that, for instance,
M
we will also write RicM
f and BR . The Riemannian m-dimensional volume will be indicated
with Vol, and the measure with density by dνf “ e´f dVol. While will write Hm´1 for the
induced pm ´ 1q-dimensional Hausdorff measure and dHfm´1 “ e´f dHm´1 . We will use the
symbol tΩj u Ò M for indicate a family tΩj ujPN of relativity compact, open sets with smooth
boundary and satisfying
`8
ď
Ωj Ť Ωj`1 Ť M, M “
Ωj ,
j“0

where A Ť B means A Ď B. Such a family will be called an exhaustion of M . Hereafter, we
consider
g P C 8 pRq,
and a solution u on M of
∆f u ` gpuq “ 0

on M.

(6.5)

We recall that u is characterized, on each open subset U Ť M , as a critical point of the
energy functional Ef : Wc1,2 pM q Ñ R given by
ż
ż
żt
1
2
Ef pwq “
|∇w| dνf ´
Gpwqdνf , where Gptq “ gpsqds,
(6.6)
2 M
M
0
with respect to compactly variation in U . Let Jf the Jacob operator of Ef at u, that is,
Jf φ “ ´∆f φ ´ g 1 puqφ,

@ φ P Cc8 pM q,

where Cc8 pM q is the space of the smooth functions compactly supported in M .
59

(6.7)

Definition 3 The function u solving p6.5q is said to be a stable solution if Jf is non-negative
on Cc8 pM q, that is, if pφ, Jf φqL2 ě 0, for all φ P Cc8 pM q. In other words,
ż
ż
1
2
(6.8)
g puqφ dνf ď
|∇φ|2 dνf ,
for all φ P Cc8 pM q.
M

M

By density, we can replace Cc8 pM q in p6.8q with Lipc pM q. By a simple adaptation of the
[21, Theorem 1], the stability of u turns out to be equivalent to the existence of a positive
w P C 8 pM q solving ∆f w ` g 1 puqw “ 0 on M .
Let Ω be an open set on M and K be a compact set in Ω. We call the pair pK, Ωq of a
f -capacitor and define the f -capacity capf pK, Ωq by
ż
capf pK, Ωq “ inf
|∇φ|2 dνf ,
(6.9)
φ P LpK,Ωq Ω

where LpK, Ωq is a set of Lipschitz functions φ on M with a compact support in Ω such that
0 ď φ ď 1 and φ|K “ 1.
For an open precompact set K Ă Ω, we define its f -capacity by
capf pK, Ωq :“ capf pK, Ωq.
In case that Ω “ M , we write capf pKq for capf pK, Ωq. It is obvious from the definition
that the set LpK, Ωq increases on expansion of Ω (and on shrinking of K). Therefore, the
capacity capf pK, Ωq decreases on expanding of Ω (and on shrinking of K). In particular, one
can prove that, for any exhaustion sequence tEk u
capf pKq :“ lim capf pK, Ek q.
kÑ8

Definition 4 A weighted Riemnnian manifold is f -parabolic if there exists no non-constant
bounded below f -superharmonic function u, that is, if ∆f u ď 0 and u ě k, for some k P R,
then u is constant.
Hence, we have the following characterization of f -parabolicity. For the proof see [25].
Proposition 6.2 Let M be a complete weighted Riemannian manifold. Then, the following
are equivalent:
1. M is f -parabolic.
2. capf pKq “ 0 for some (then any) compact set K Ă M .
The following criterion of f -parabolicity is well known, for more details see for instance
[22, Proposition 3.4].
Proposition 6.3 Let po be a fixed point in a weighted Riemannian manifold M and let
ż
ż
m´1
Lprq “
dHf
and
V prq “
dνf .
BBppo ,rq

Bppo ,rq

If
ż8
1

dr
“ `8
Lprq

ż8
or
1

then M is f -parabolic.
60

rdr
“ `8,
V prq

6.3

Technical Computations

We start this section with a Picone type identity in a weighted Riemannian manifold.
Lemma 2 Let Ω Ď M be a domain with C 3 boundary ppossibly emptyq and let u P C 3 pΩq be
a solution of ´∆f u “ gpuq on Ω. Let w P C 1 pΩq X C 2 pΩq be a solution of ∆f w ` g 1 puqw ď 0
such that w ą 0 on Ω. Then the following inequality holds true: for every ε ą 0 and for
every φ P Lipc pM q,
ż
ż
ż
w
φ2
2
m´1
ď |∇φ| dνf ´ g 1 puq
pBν wqdHf
φ2 dνf
(6.10)
w
`
ε
w
`
ε
Ω
Ω
BΩ
ˇ ˆ
˙ˇ2
ż
ˇ
ˇ
φ
2ˇ
ˇ dνf .
´ pw ` εq ˇ∇
ˇ
w
`
ε
Ω

Furthermore, if either Ω “ M or w ą 0 on Ω, one can also take ε “ 0 inside the above
inequality. The inequality is indeed an equality if w solves ∆f w ` g 1 puqw “ 0 on Ω.
Proof. We integrate ∆f w ` g 1 puqw ď 0 against the test function φ2 {pw ` εq to deduce
ż
ż
φ2
φ2
1
0 ď ´ p∆f w ` g puqwq
dνf “ ´
pBν wqdHfm´1
(6.11)
w
`
ε
w
`
ε
Ω
BΩ
F
ż B ˆ 2 ˙
ż 1
φ
g puqwφ2
`
∇
, ∇w dνf ´
dνf
w`ε
Ω
Ω w`ε

Since

B

ˆ
∇

φ2
w`ε

˙

F
, ∇w

“2

φ
φ2
x∇φ, ∇wy ´
|∇w|2 ,
w`ε
pw ` εq2

using the identity
ˇ ˆ
˙ˇ2
ˇ
ˇ
φ
φ2
φ
ˇ “ |∇φ|2 `
pw ` εq ˇ∇
|∇w|2 ´ 2
x∇w, ∇φy,
ˇ
2
w`ε
pw ` εq
w`ε
2ˇ

we infer that

ˇ ˆ
B ˆ 2 ˙
F
˙ˇ2
ˇ
ˇ
φ
φ
2
2ˇ
ˇ .
∇
, ∇w “ |∇φ| ´ pw ` εq ˇ∇
w`ε
w`ε ˇ

Inserting p6.12q into p6.11q we get the desired p6.10q.

61

(6.12)

Proposition 6.4 In the above assumptions, for every ε ą 0 the following integral inequality
holds true:
ż
ż
φ2 w
2
r|Hessu| ` Ricf p∇u, ∇uqs
dνf ´ φ2 |∇|∇u||2 dνf ď
(6.13)
w
`
ε
Ω
Ω
„
ˆ
˙

ż
|∇u|2
φ2
2
wBν
´ |∇u| Bν w dHfm´1 `
ď
w
`
ε
2
BΩ
˙F
B
ˆ
ż
ż
φ
1
w
2
2
2
`ε
x∇φ, ∇|∇u| ydνf ´
dνf `
φ ∇|∇u| , ∇
2 Ω
w`ε
Ω w`ε
ˇ ˆ
˙ˇ2
ż
ż
ˇ
φ|∇u| ˇˇ
2
2
2ˇ
dνf .
` |∇φ| |∇u| dνf ´ pw ` εq ˇ∇
w`ε ˇ
Ω
Ω

Furthermore, if either Ω “ M or w ą 0 on Ω, one can also take ε “ 0. The inequality is
indeed an equality if ∆f w ` g 1 puqw “ 0 on Ω.
Proof. We start with the B:ochner formula
1
∆f |∇u|2 “ |Hessu|2 ` x∇u, ∇p∆f uqy ` Ricf p∇u, ∇uq,
2
valid for each u P C 3 pΩq. Since u solves ´∆f u “ gpuq, we get
1
∆f |∇u|2 “ |Hessu|2 ´ g 1 puq|∇u|2 ` Ricf p∇u, ∇uq.
2
Integrating p6.14q on Ω against the test function ψ “ φ2 w{pw ` εq we deduce
ż
r|Hessu|2 ` Ricf p∇u, ∇uqsψ dνf “
Ω
ż
ż
2
1
φ2 w
1
2 φ w
dνf `
∆f |∇u|2 dνf
“ g puq|∇u|
pw ` εq
2 Ω pw ` εq
żΩ
ż
2
1
wφ2
1
2 φ w
“ g puq|∇u|
dνf `
Bν |∇u|2 dHfm´1
pw
`
εq
2
w
`
ε
Ω
BΩ
˙
F
ż B ˆ
1
wφ2
2
´
∇
, ∇|∇u| dνf
2 Ω
w`ε
ż
ż
2
1
wφ2
1
2 φ w
“ g puq|∇u|
dνf `
Bν |∇u|2 dHfm´1
pw
`
εq
2
w
`
ε
Ω
BΩ
ż
wφ
´
x∇φ, ∇|∇u|2 ydνf
w
`
ε
Ω
B
ˆ
˙F
ż
1
w
2
2
´
φ ∇|∇u| , ∇
dνf .
2 Ω
w`ε

62

(6.14)

(6.15)

Next, we consider the inequality p6.10q with the test function φ|∇u| P Lipc pM q:
ż
ż
ż
2
w
2
m´1
2 φ
ď |∇pφ|∇u|q| dνf ´ g 1 puq
pBν wqdHf
φ2 |∇u|2 dνf
|∇u|
w
`
ε
w
`
ε
Ω
BΩ
ˇ ˆ Ω
˙ˇ2
ż
ˇ
φ|∇u| ˇˇ
2ˇ
dνf
´ pw ` εq ˇ∇
w`ε ˇ
Ω
ż
ż
2
2
“ |∇φ| |∇u| dνf ` φ2 |∇|∇u||2 dνf
Ω ż
Ω
` 2 φ|∇u|x∇φ, ∇|∇u|ydνf
Ω
ż
w
´ g 1 puq
|∇φ|2 φ2 dνf
w
`
ε
Ω
ˇ ˆ
˙ˇ2
ż
ˇ
φ|∇u| ˇˇ
2ˇ
´ pw ` εq ˇ∇
dνf .
w`ε ˇ
Ω

(6.16)

Recalling that ∇|∇u|2 “ 2|∇u|∇|∇u| weakly on M , summing up 6.15 and p6.16q, putting
together the terms of the same kind and rearranging we deduce p6.13q as desired.
Corollary 6.1 In the above assumptions, if it holds
ˆ
˙
ż
w
2
2
φ x∇|∇u| , ∇
ydνf ě 0.
lim inf
εÑ0`
w`ε
Ω

(6.17)

Then
ż
“

‰
|Hessu|2 ` Ricf p∇u, ∇uq ´ |∇|∇u||2 φ2 dνf `
Ω
ˇ ˆ
˙ˇ2
ż
ż
ˇ
φ|∇u| ˇˇ
2ˇ
2
2
pw ` εq ˇ∇
` lim inf
ˇ dνf ď |∇φ| |∇u| dνf `
εÑ0`
w
`
ε
Ω
„
ˆ
˙

żΩ
2
2
φ
|∇u|
2
` lim inf
wBν
´ |∇u| Bν w dHfm´1 .
`
εÑ0
2
BΩ w ` ε

(6.18)

Proof. It is an immediate consequence of the Proposition 6.4.
The next result is known in the literature, we present it here for sake of completeness.
Proposition 6.5 Let u P C 2 pM q, and let p P M be a point such that ∇uppq ‰ 0. Then,
denoting with |A|2 the square norm of second fundamental form of the level set Σ “ tu “ uppqu
in a neighborhood of p, it holds
|Hessu|2 ´ |∇|∇u||2 “ |∇u|2 |A|2 ` |∇T |∇u||2 ,
where ∇T is the tangential gradient on the level set Σ.
Proof. Fix a local orthonormal frame tei u on Σ, and let ν “ ∇u{|∇u| be the normal vector.
For every vector field X P XpM q,
Hessupν, Xq “

1
1
Hessup∇u, Xq “
x∇|∇u|2 , Xy “ x∇|∇u|, Xy.
|∇u|
2|∇u|
63

Moreover, for a level set
A“´

Hessu|T ΣˆT Σ
,
|∇u|

we have
|Hessu|2 “

ÿ

pHessupei , ej qq2 ` 2

i,j

ÿ

pHessupν, ej qq2 ` pHessupν, νqq2

j

“ |∇u|2 |A|2 ` 2

ÿ

x∇|∇u|, ej y2 ` x∇|∇u|, νy2

j

“ |∇u|2 |A|2 ` |∇T |∇u||2 ` |∇|∇u||2 ,
proving the proposition.

6.4

Proof of the Theorem 6.1

Now we are ready to prove our main theorem.
Proof of Theorem 6.1. In our assumption, we consider the integral formula p6.13q with
Ω “ M and ε “ 0. Since Ricf ě 0 we deduce
ż

ż
“

2

2

‰

ż

2

2

|Hessu| ´ |∇|∇u|| φ dνf ď

M

2

ˇ ˆ
˙ˇ2
ˇ
φ|∇u| ˇˇ
w ˇ∇ ∇
ˇ dνf .
w
2ˇ

|∇φ| |∇u| dνf ´
M

M

(6.19)

Next, we rearrange the right hand side as follows: using the inequality
|X ` Y |2 ě |X|2 ` |Y |2 ´ 2|X||Y | ě p1 ´ δq|X|2 ` p1 ´ δ ´1 q|Y |2 ,
valid for each δ ą 0, we obtain
ˇ
ˇ ˆ
˙ˇ2
˙ˇ2
ˆ
ˇ
ˇ
|∇u| ˇˇ
φ|∇u| ˇˇ
2 ˇ |∇u|∇φ
2ˇ
` φ∇
w ˇ∇
ˇ “w ˇ w
w
w ˇ

(6.20)

ˇ ˆ
˙ˇ2
ˇ
|∇u| ˇˇ
.
ě p1 ´ δ q|∇u| |∇φ| ` p1 ´ δqφ w ˇ∇
w ˇ
´1

2

2

2

2ˇ

(6.21)

Substituting in p6.19q yields
ˇ ˆ
˙ˇ2
ˇ
|∇u| ˇˇ
|Hessu| ´ |∇|∇u|| φ dνf ` p1 ´ δq
φ w ˇ∇
dνf
w ˇ
M
M
ż
1
ď
|∇φ|2 |∇u|2 dνf .
δ M

ż

ż

“

2

2

‰

2

2

2ˇ

Choose δ ă 1. We claim that, for suitable families tφα uαPIĎR` , it holds
ż
tφα u is monotone increasing to 1, lim
|∇φα |2 |∇u|2 dνf “ 0.
αÑ`8 M

64

(6.22)

Choose φ as follows, according to the case.
In the case (i), fix Ω Ť M with smooth boundary and let tΩj u Ò M be a smooth
exhaustion with Ω Ă Ω1 . Choose φ “ φj P Lip0 pM q to be identity 1 on Ω, 0 on M zΩj and
the f -harmonic capacitor on Ωj zΩ, that is, the solution of
$
’
on Ωj zΩ
&∆f φj “ 0
φj “ 1
on BΩ,
’
%
φj “ 0
on BΩj .
By comparison and since M is f -parabolic, tφj u is monotonically increasing and pointwise
convergent to 1, and moreover
ż
|∇φj |2 |∇u|2 dνf ď |∇u|2L8 capf pΩ, Ωj q Ñ |∇u|2L8 capf pΩq “ 0,
Ωj

the last equality follows of Proposition 6.2 since M is f -parabolic. This proves p6.22q.
In the case (ii), we apply a logarithmic cut-off argument. For fixed R ą 0, choose the
following radial function φpxq “ φR prpxqq:
$
?
’
if r ď R,
&1
?
log r
φR prq “ 2 ´ 2 log
(6.23)
if r P r R, Rs,
R
’
%
0
if r ě R.
Note that
|∇φpxq|2 “

4
rpxq2 log2 R

χBR zB?R pxq,

where χA is the characteristic function of a subset A Ď M . Choose R in such a ways that
log R{2 is an integer. Then
ż
ż
2
2
|∇φ| |∇u| dνf “
|∇φ|2 |∇u|2 dνf
(6.24)
BR zB?R

M

logÿ
R´1 ż
4
|∇u|2
“
dνf
log2 R k“log R{2 Bek`1 zBek rpxq2
log
ÿR 1 ż
4
ď
|∇u|2 dνf .
log2 R k“log R{2 e2k Bek`1

By assumption

ż
|∇u|2 dνf ď pk ` 1qe2pk`1q δpkq
Bek`1

for some δpkq satisfying δpkq Ñ 0 as k Ñ `8. Without loss of generality, we can assume
δpkq to be decreasing as a function of k. Whence,

65

log
log
ÿR 1 ż
ÿR e2pk`1q
4
8
2
|∇u| dρ ď
pk ` 1qδpkq
log2 R k“log R{2 e2k Bek`1
log2 R k“log R{2 e2k

(6.25)

log
ÿR
8e2
δplog R{2q
pk ` 1q
ď
log2 R
k“0
C
ď
δplog R{2q log2 R
2
log R
“ Cδplog R{2q,

for some constant C ą 0. Combining p6.24q and 6.25 and letting R Ñ `8 we deduce p6.22q.
Therefore, in both the cases, we can infer from the integral formula (6.19) that
|∇u| “ cw, for some c ě 0, |Hessu|2 “ |∇|∇u||2 , Ricf p∇u, ∇uq “ 0.

(6.26)

Since u is non-constant by assumption, we have c ą 0 and thus |∇u| ą 0 on M . From
B:ochner formula, it holds
1
|∇u|∆f |∇u| ` |∇|∇u||2 “ ∆f |∇u|2 “ |Hessu|2 ´ g 1 puq|∇u|2 ` Ricf p∇u, ∇uq
2
on M . Using p6.26q, we obtain that ∆f |∇u| ` g 1 puq|∇u| “ 0 on M , hence |∇u| (and so w)
both solve the linearized equation Jf v “ 0.
Now, the flow Φ of ν “ ∇u{|∇u| is well defined on M . Since M is complete and |ν| “ 1,
Φ is defined on M ˆ R. By p6.26q and Proposition 6.5, |∇u| is constant on each connected
component of a level set N , and N is totally geodesic. Therefore, in a local Darboux frame
tej , νu for the level surface N , we have that
"
0 “ |II|2 implies Hessupei , ej q “ 0,
(6.27)
0 “ x∇|∇u|, ej y “ Hessupν, ej q,
so the unique component of Hessu is that corresponding to the pair pν, νq. Now we will prove
that γ is a geodesic. Indeed, let X P XpM q be a vector field, we have that
ˆ
˙
∇u
1
1
x∇γ 1 γ , Xy “
x∇∇u
, Xy
|∇u|
|∇u|
1
1
“
x∇∇u ∇u, Xy ´
x∇up|∇u|q∇u, Xy
2
|∇u|
|∇u|3
1
1
Hessup∇u, Xq ´
x∇|∇u|, ∇uyx∇u, Xy
“
2
|∇u|
|∇u|3
1
1
“
Hessupν, Xq ´
x∇|∇u|, νyxν, Xy
|∇u|
|∇u|
1
1
“
Hessupν, Xq ´
Hessupν, νqxν, Xy “ 0,
|∇u|
|∇u|
where the last line follows from p6.27q. So, ∇γ 1 γ 1 “ 0 and γ is a geodesic as desired.
66

Following the arguments in the proof of [40, Theorem 9.3], we will prove the topological
splitting. Since |∇u| is constant on level sets of u, |∇u| “ βpuq for some function β. Evaluating along curves Φt pxq, since u ˝ Φt is a local bijection we deduce that β is continuous.
Claim 6.6 Φt moves level sets of u to level sets of u.
d
pu ˝ Φs q “ |∇u| ˝ Φs “ βpu ˝ Φs q we get
Indeed, integrating ds

ż upΦt pxqq
t“
upxq

dξ
,
βpξq

thus upΦt pxqq is independent of x varying in a level set. As βpξq ą 0, this also show that
flow lines starting from a level set of u do not touch the same level set and we conclude the
Claim.
Let N be a connected component of a level set of u.
Claim 6.7 Φ|N ˆR is surjective.
In fact, since the flow of ν is through geodesics, for each x P N , Φt coincides with the normal
exponential map expK ptνpxqq. Moreover, since N is closed in M and M is complete, the
normal exponential map is surjective because each geodesic from x P M to N minimizing
distpx, N q is perpendicular to N (by variational arguments).
Claim 6.8 Φ|N ˆR is injective.
Suppose that Φpx1 , t1 q “ Φpx2 , t2 q. Then, since Φ moves level sets to level sets, necessarily
t1 “ t2 “ t. If by contradiction x1 ‰ x2 , two distinct flow lines of Φt would intersect at the
point Φt px1 q “ Φt px2 q, contradicting the fact that Φt is a diffeomorphism on M for every t,
as desired.
Thus, we conclude that Φ : N ˆ R Ñ M is a diffeomorphism. In particular, each level set
Φt pN q is connected. This proves the topological part of the splitting.
To conclude the splitting, we will prove that Φt is an isometry for all t, that is, we will
prove that ν is a Killing vector field. Indeed, we consider the Lie derivative of the metric in
the direction of ν:
pLν gM qpX, Y q “ x∇X ν, Y y ` xX, ∇Y νy
ˆ
˙
ˆ
˙
2
1
1
HessupX, Y q ` X
x∇u, Y y ` Y
x∇u, Xy.
“
|∇u|
|∇u|
|∇u|
From the expression, using that |∇u| is constant on N and the properties of Hessu we deduce
that
2
pLν gM qpX, Y q “
HessupX, Y q “ 0,
|∇u|

67

if at least one between X and Y is in the tangent space of N . If, however, X and Y are
normal (w.l.o.g. X “ Y “ ∇u), we have
ˆ
˙
1
2
Hessup∇u, ∇uq ` 2∇u
|∇u|2
pLν gM qpX, Y q “
|∇u|
|∇u|
2
“
Hessup∇u, ∇uq ´ 2∇up|∇u|q
|∇u|
“ 2Hessupν, ∇uq ´ 2x∇|∇u|, ∇uy “ 0.
Thus, we conclude that ν is a Killing field and so Φt is a flow of isometries. Since
∇u K T N, M splits as a Riemannian product, as desired. In particular, RicN
f ě 0 if m ě 3,
while, if m “ 2, M “ R2 or S1 ˆ R with the flat metric.
Lastly, we will verify the properties of the function u. Let γ be any integral curve of ν.
Then
d
pu ˝ γq “ x∇u, νy “ |∇u| ˝ γ ą 0,
dt
since |∇u| ą 0. Now, as M splits isometrically in the direction of ∇u we obtain that
Ricpν, νq “ 0 and this implies that Hessf pν, νq “ 0. Consequently x∇f, νy “ k is constant in
the splitting direction.
By the other hand,
´gpu ˝ γq “ ∆f upγq “ Hessupν, νqpγq ´ x∇f, ∇uypγq
“ x∇|∇u|, νypγq ´ x∇f, νy|∇u|pγq
d
“ p|∇u| ˝ γq ´ k|∇u| ˝ pγq
dt
d2
d
“ 2 pu ˝ γq ´ k pu ˝ γq,
dt
dt
2
1
and thus y “ u ˝ γ solves the ODE ´y ` ky “ gpyq with y 1 ą 0.
We next address the parabolicity. Under assumption (i), M is f -parabolic and so N is
necessarily f -parabolic too. We are going to deduce the same under assumption (ii). Note
that the chain of inequalities
ˆż R
˙
ż
1
2
N
|y ptq| dt Volf pBR q ď
|y 1 ptq|2 dt dνfN
N
´R
r´R,RsˆBR
ż
|∇u|2 dνf “ opR2 log Rq
ď
BR?2

gives immediately p6.3q and p6.4q, since |y 1 | ą 0 everywhere. Thus, since Volf pBRN q “
opR2 log Rq, we know that there is a constant A such that Volf pBRN q ď AR2 log R, that
is,
R
1
ě
,
N
AR log R
Volf pBR q
hence
żt
żt
R dR
dR
lim
ě lim
N
tÑ8 1 Volf pB q
tÑ8 1 AR log R
R
“ A´1 lim logplog tq “ 8.
tÑ8

68

Thus, by proposition 6.3, N is f -parabolic. So we conclude the proof.

69

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