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                    Universidade Federal de Alagoas – UFAL
Instituto de Matemática – IM
Plano de Atividades Acadêmicas 2020.2

Docente: José Anderson de Lima e Silva
SIAPE: 3140413
Atividades de Ensino:
Atividades de orientação, supervisão e outros em nível de Graduação e PósGraduação.
1. Atendimento extraclasse de turmas com mais de 40 alunos (2 turmas). 3 pontos;
2. Monitoria (4 supervisões). 4 pontos;
3. Iniciação científica (1 supervisão). 2 pontos.
Atividades de Pesquisa:
Participação em projetos de pesquisa que tenham entre suas metas a divulgação
dos resultados conforme Art. 10º, Incisos I, II, III. 20 pontos.
1. Coordenador do Projeto de Pesquisa: Teoria Min-max aplicada as Superfícies
Mínimas.
Escrita de artigos:



The widths of the riemannian product of a unit circle with a unit two-sphere;
Low min-max widths of the 3-dimensional real projective space.

Obs.: Continuação do projeto aprovado pelo PPGMAT. Durante este projeto já
escrevemos dois preprints que estão submetidos: “A SHORT NOTE ABOUT 1WIDTH OF LENS SPACES” e “THE FIRST AND SECOND WIDTH OF THE REAL
PROJECTIVE SPACE”.
Atividades Administrativas e de Representação:
1. Coordenação de monitoria. 6 pontos;
2. Participação em reunião de colegiado ou CONSIM. 1 ponto.

Pontuação total: 36 pontos.
Pontuação do PAA: 24 pontos.

A SHORT NOTE ABOUT 1-WIDTH OF LENS SPACES
MÁRCIO BATISTA∗ AND ANDERSON DE LIMA
Abstract. In this note we explore the nature of Lens spaces to
study the first width of those spaces, more precisely we use the
existence of a sharp sweepout associated to a Clifford torus to
provide a simple and pretty application of the Willmore conjecture
for the computation of the 1-width of Lens Spaces.

1. Introduction
In a remarkable work, [2], Almgren proved that:
The width of the round 3-sphere is equal to 4π and the surface which
reaches this number is an equatorial sphere.
Later on, using, in a fundamental way, the symmetric structure of
the round unit sphere in R4 , Nurser motived by a question proposed
by Marques & Neves, [14], computed the values of some widths of the
round unit sphere, more precisely, he computed
ω1 (S 3 ) = ω2 (S 3 ) = ω3 (S 3 ) = ω4 (S 3 ) = 4π and ω5 (S 3 ) = 2π 2 ,
and he also gets some bounds to ω9 and ω13 , see [16].
From the close relationship between Sn and RPn , do Carmo, Ros
and Ritoré studied closed minimal hypersurfaces in RPn and classified
those hypersurfaces with index one, see [5]. More recently, RamirezLuna gets a similar result for hypersurfaces immersed in CPn and using
this classification and some tools from the min-max theory, she gets the
value of the width of RPn for n between 3 and 7, see [18]. Using another
approach, the authors get the value of the first and second widths of
RPn for n in the same range as before, see [4].
Motivated by questions in the works of Guth [9], Gromov [8] and
Marques & Neves [14], others works were done and the value of widths
were computed, see for instance [1], [16], [3] and [6]. These works used
strongly the symmetric structure of the target spaces.
2010 Mathematics Subject Classification. Primary: 53C42; Secondary: 49J35.
Key words and phrases. Minimal surfaces, p-width, Min-max theory.
∗
Corresponding author.
1

THE FIRST AND SECOND WIDTH OF THE REAL PROJECTIVE SPACE
MÁRCIO BATISTA∗ AND ANDERSON DE LIMA
Abstract. In this paper we deal with the first and second width of the real projective space
RPn , for n in the range 4 to 7, in the setting of Almgren-Pitts min-max theory. In a recent
paper, Ramirez-Luna, using a result due to do Carmo, Ritoré and Ros, computed the first width
of the real projective spaces and, at the same time, we obtain optimal sweepouts which realize
the first and second widths of those spaces.

1. Introduction
Liokumovich, Marques and Neves in [8] were interested in understand the volume spectrum
of a given Riemannian manifold, {ωp (M )}p , and they were able to prove a Weyl law for such
non-linear spectral problem as conjectured by Gromov in [5] and so they obtained the more
quantitative result after the works done by Gromov [5] and Guth [6]. Using also the volume
spectrum, Marques and Neves provided a proof of one famous conjecture of Yau about minimal
surfaces in the setting of Frankel property and in the same work, see [10, Section 9], they point
out that would be interesting to compute the values of ωp (M ) in specific examples and verify
whether such numbers are achieved by interesting minimal hypersurfaces. Therefore, in a recent
work, see [2], the authors computed and provided some upper bounds for the widths on the real
projective 3-space. More precisely, the authors proved:
For the p-widths of the 3-dimensional real projective space, RP3 ,
we have
• ω1 (RP3 ) = ω2 (RP3 ) = ω3 (RP3 ) = π 2 ;
• ω9 (RP3 ) ≤ 4π,
and the surface which realizes this number π 2 is the minimal Clifford torus T 2 /{−x, x} in RP3 . Moreover, there exists a closed
minimal surface Σ, with genus greater or equal to two, realizing
the 4-width.
The computations of the widths in the above result relies on a number of crucial ingredients:
the results of Almgren and Pitts [13], [16] and [7] about existence of closed embedded minimal
hypersurfaces with index one and a construction of an optimal sweepouts. Moreover, the authors
used an algebraic sweepout to provide an upper bound to the ninth width.
We note that the interest about the asymptotic behavior of the volume spectrum appears
first in [5] and [6], and later on in [8]. In the first two works cited, the authors obtain an upper
and lower bound of the p-width as a particular power of its parameter p. In [10], Marques and
Neves developed the min-max theory and using the p-widths they proved a famous conjecture
of Yau, as noted before.
We highlight that produce an optimal sweepout and so compute the width is a hard task and
we have just a little works with this purpose. As example we cite one of the breakthroughs done
by Marques and Neves, [9]. Roughly speaking, after them developed the necessary machinery,
they introduced an optimal 5-sweepout associated to the minimal Clifford torus immersed in the
2010 Mathematics Subject Classification. Primary: 53C42; Secondary: 49J35.
Key words and phrases. Clifford hypersurfaces, Minimal hypersurfaces, Second width, Min-max theory.
∗
Corresponding author.
1