Title: Concentration of mean exit times

Abstract: The mean exit time function, when defined on a narrow tubular neighborhood (called a delta-tube) around any equator of the sphere, increases without bound as the dimension increases. In contrast, when the same function is defined on geodesic balls, it tends to zero as the dimension grows. This shows that the mean exit time exhibits a kind of concentration phenomenon, often referred to as the fat equator effect, in the case of equatorial tubes.

A similar concentration behavior is observed when the mean exit time function is considered on tubes around closed, minimal hypersurfaces of a compact Riemannian manifold with Ricci curvature bounded below by (n−1). In this setting, a Brownian particle starting its random movement near such a hypersurface will remain close to it for increasingly longer periods as the dimension of the ambient manifold increases, eventually spending an infinite amount of time near the hypersurface in the limit.

Title: Rigidity of compact quasi-Einstein manifolds with boundary

Abstract:  It is known by the classical book "Einstein Manifolds" (Besse, 1984) that quasi-Einsteinmanifolds 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 S^3_{+}, or the cylinder IxS^2 with product metric. For dimension n=4, we will show that a 4-dimensional simply connected compact quasi-Einsteinmanifold with boundary and constant scalar curvature is isometric to either the standard hemisphere S^4_+, or the cylinder I x S^3 with product metric, or the product space S^2_+ x S^2 with the product metric. This is a joint work with D. Zhou and E. Ribeiro Jr.