Gap theorems for free-boundary submanifolds

Início: 25/06/2020 17:00

Término: 25/06/2020 18:00

Palestrante: Marcos Petrucio Cavalcante (Universidade Federal de Alagoas)

E-mail do Palestrante:

Resumo: Let $M^{n}$ be a compact $n$ -dimensional manifold minimally immersed in a unit sphere $S^{n + k}$ and let denote by $| A |^{2}$ the squared norm of its second fundamental form. It follows from the famous Simons pinching theorem that if $| A |^{2} l e q f r a c n 2 - f r a c 1 k$ , then either $| A |^{2} = 0$ or $| A |^{2} = f r a c n 2 - f r a c 1 k$ . The submanifolds on which $| A |^{2} = f r a c n 2 - f r a c 1 k$ were characterized by Lawson (when $k = 1$ ) and by Chern-do Carmo-Kobayashi (for any $k$ ). These important results say that there exists a gap in the space of minimal submanifolds in $S^{n + k}$ in terms of the length of their second fundamental forms and their dimensions. Latter, Lawson and Simons proved a topological gap result without making any assumption on the mean curvature of the submanifold. Namely, they proved that if $M^{n}$ is a compact submanifold in $S^{n + k}$ such that $| A |^{2} l e q m i n p (n - p), 2 s q r t p (n - p)$ , then for any finitely generated Abelian group $G$ , $H_{p} (M; G) = 0$ . In particular, if $| A |^{2} < m i n n - 1, 2 s q r t n - 1$ , then $M$ is a homotopy sphere. It is well known that free-boundary minimal submanifolds in the unit ball share similar properties as compact minimal submanifolds in the round sphere. For instance, Ambrozio and Nunes obtained a geometric gap type theorem for free-boundary minimal surfaces $M$ in the Euclidean unit $3$ -ball $B^{3}$ . They proved that if $| A |^{2} (x) l a n g l e x, N (x) r a n g l e^{2} l e q 2$ , where $N (x)$ is the unit normal vector at $x i n M$ , then $M$ is either the equatorial disk or the critical catenoid. In the first part of this talk, I will present a generalization of Ambrozio and Nunes theorem for constant mean curvature surfaces. Precisely, if the traceless second fundamental form $p h i$ of a free-boundary CMC surface $B^{3}$ satisfies $| p h i |^{2} (x) l a n g l e x, N (x) r a n g l e^{2} l e q (2 + H l a n g l e x, N (x) r a n g l e)^{2} / 2$ then $M$ is either a spherical cap or a portion of a Delaunay surface. This is joint work with Barbosa and Pereira. In the second part, I will present a topological gap theorem for free-boundary submanifolds in the unit ball. More precisely, if $| p h i |^{2} l e q f r a c n p n - p$ , then the $p$ -th cohomology group of $M$ with real coefficients vanishes. In particular, if $| p h i |^{2} l e q f r a c n n - 1$ , then $M$ has only one boundary component. This is joint work with Mendes and Vitório.