# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### The Yamabe problem for the conformal fractional Laplacian

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type maximum principle together with interplays between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extend the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem. This is joint work with J. Qing.

##### Parabolic equations and the Ricci flow on manifolds with boundary

In the first part of the talk, we will focus on a second-order quasilinear parabolic equation in a vector bundle over a compact manifold~$M$ with boundary. Our goal is to explore the short-time existence of solutions to this equation. In the second part, we will discuss the Ricci flow on~$M$. The objective is to propose a new boundary condition for the flow and state two short-time existence results.

##### Geometry of level sets of elliptic equations and a conjecture of De Giorgi

I will describe recent results on the rigidity of level sets of solutions of local and non local elliptic equations on the euclidean space and on riemannian manifolds in connection with a conjecture by De Giorgi.