Conical metrics of prescribed Gaussian and geodesic curvatures on compact surfaces
偏微分方程几何分析
报告题目(Title)：Conical metrics of prescribed Gaussian and geodesic curvatures on compact surfaces
报告人(Speaker)：Mohameden Ahmedou（Giessen University，Germany)
地点(Place)：后主楼1223
时间(Time)：2019年7月4日 16:00-17:00
邀请人(Inviter)：唐仲伟
报告摘要
We consider the problem of finding conformal metrics with prescribed Gauss curvature and zero geodesic curvature. This amounts to solve a nonlinear Liouville equation under Neumann boundary condition and hence enjoys a variational structure. Moreover it turns out that, as far as the variational aspects are concerned, one has to distinguish between the "resonant" and "non resonant" case depending on whether or not the sum of the integrals of the Gauss curvature on the surfaces and the integral of the geodesic curvature on the boundary takes some explicit critical values. Indeed in the "non resonant" the associated variational problem is compact, while it is non compact in the resonant one. Using a Morse theoretical approach, we prove some existence results in the "non resonant " case and establish Morse Inequalities. A major role in our argument is played by some "boundary-weighted barycentric" set. In the resonant one we study the "critical points at Infinity" and derive some Euler-Poincaré type criterion of the existence of solutions. This is a joint work with Thomas Bartsch(Giessen) and Marcello Lucia (CUNY, New York)
