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 $|A|^2leq frac{n}{2-frac{1}{k}}$, then either $|A|^2=0$ or $|A|^2=frac{n}{2-frac{1}{k}}$. The submanifolds on which $|A|^2=frac{n}{2-frac{1}{k}}$ 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 $|A|^2leq min{p(n-p), 2sqrt{p(n-p)}}$, then for any finitely generated Abelian group $G$, $H_p(M;G)=0$. In particular, if $|A|^2< min{n-1, 2sqrt{n-1}}$, 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 $|A|^2(x)langle x, N(x)rangle^2leq 2$, where $N(x)$ is the unit normal vector at $xin M$, 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 $phi$ of a free-boundary CMC surface $B^3$ satisfies $|phi|^2(x)langle x, N(x)rangle^2leq (2+Hlangle x, N(x)rangle )^2/2$ 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 $|phi|^2leq frac{np}{n-p}$, then the $p$-th cohomology group of $M$ with real coefficients vanishes. In particular, if $|phi|^2leq frac{n}{n-1}$, then $M$ has only one boundary component. This is joint work with Mendes and Vitório.