Topology and stability index of minimal hypersurfaces and self-shrinkers
报告题目(Title)：Topology and stability index of minimal hypersurfaces and self-shrinkers
报告人(Speaker)：Alessandro Savo (Sapienza Universita di Roma, Italy)
地点(Place)：后主楼1220
时间(Time)：6月11日上午9:00-10:00
邀请人(Inviter)：彦文娇
报告摘要
It is well-known that the first variation of the area functional of a given hypersurface of a Riemannian manifold is zero for all variation vector fields if and only if the hypersurface is minimal. The second variation is a quadratic form whose number of negative eigenvalues is often called the Morse (or stability) index of the given minimal immersion: it measures, in a sense, the number of independent directions along which the area is decreasing. A conjecture by Schoen, Marques and Neves states that, for ambient manifolds of positive Ricci curvature, the stability index is bounded below by a linear function of the first Betti number; in particular, a large de Rham cohomology space in degree one implies high instability. In the talk we review the main results in this direction and discuss a recent generalization to weighted minimal hypersurfaces.
