GeCo GeDi Seminarshttps://gecogedi.dimai.unifi.it/seminars/en-usFri, 11 Oct 2024 20:49:52 +0000Rigidity of compact quasi-Einstein manifolds with boundaryhttps://gecogedi.dimai.unifi.it/seminar/1656/2024-10-14: E. Ribeiro de Sousa JĂșnior.<p>It is known by the classical book "Einstein Manifolds" (Besse, 1984) that quasi-Einstein manifolds correspond to a base of a warped product Einstein metric. Another interesting motivation to investigate quasi-Einstein manifolds derives from the study of diffusion operators by Bakry and Emery (1985), which is linked to the theories of smooth metric measure space, static spaces and Ricci solitons. In this talk, we will show that a 3-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature must be isometric to either the standard hemisphere S3<sub>{+},</sub> or the cylinder R x S2 with product metric. For dimension n=4, we will prove that a 4-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric to either the standard hemisphere S4<sub>+</sub> or the cylinder I x S3 with product metric, or the product space S2<sub>+</sub> x S2 with the product metric.</p>https://gecogedi.dimai.unifi.it/seminar/1656/Lower bound for the first eigenvalue of compact minimal hypersurfaces of the Euclidean spherehttps://gecogedi.dimai.unifi.it/seminar/1657/2024-10-14: A. Alves de Barros.<p>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 conjecturedby 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 $\Lambda = \sup_M <br> A <br>\ge\sqrt n$. 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.</p>https://gecogedi.dimai.unifi.it/seminar/1657/