14 oct 2024 -- 16:00
Università dell'Insubria
Abstract.
Let $x : M^n\to S^{n+1}$ be a minimal immersion of a compact Riemannian manifold $M^n$ into the Euclidean unit sphere $S^{n+1}$ with second fundamental form $A$. It is well known that $\Delta x + nx = 0$, where $\Delta$ stands for the Laplacian in the metric induced by $x$. Hence $n$ is an upper bound for the first eigenvalue $\lambda_1$ of $\Delta$. When $x(M)$ is not embedded there are examples of minimal tori with $\lambda_1 < 2$. However, when $x$ is embedded it was conjectured
by Yau that $\lambda_1= n$. The first global result in the direction of such problem was obtained by Choi and Wang around 1983, where they proved that $\lambda_1\ge\frac n2$. In a paper due to Barros and Bessa around 1999 we have improved Choi and Wang result by showing that $\lambda_1\ge\frac n2 +c(M^n,x)$, where $c(M^n,x)$ is a positive constant depending on $M^n$ and $x$. Recently, Spruck et al. as well as Zhou et al. presented a new estimate showing that $\lambda_1\ge\frac n2 + c(n, \Lambda)$, where $c(n, \lambda)$ is a small constant which depends only on $n$ and $\Lambda = \sup_M
A
\ge\sqrt n$. We point out that $\lambda_1 = n$ for the class of isoparametric hypersurfaces. The first case part was done by Muto in 1988, and, Tang and Yan completed the second part of the problem in 2015. We present a new estimate that improved the results due to Spruck et al. as well as Zhou et al. This is a joint work with D. Eliakim.