# Geometric view of conformal PDEs

# Geometric view of conformal PDEs

THIS SEMINAR HAS BEEN MOVED FROM OCTOBER 25 TO OCTOBER 18. PLEASE NOTE SPECIAL TIME. In this talk we develop a global correspondence between immersed horospherically convex hypersurfaces $\phi: {\rm M}^n\to H^{n+1}$ and complete conformal metrics $e^{2\rho}g_{S^n}$ on domains $\Omega$ in the boundary $S^n$ at infinity of $H^{n+1}$ such that $\rho$ is the horospherical support function and that $\partial_\infty\phi({\rm M}^n) = \partial\Omega$. For instance, we are able to obtain an explicit correspondence between Obata's Theorem (for conformal metrics) and Alexandrov Theorem (for hypersurfaces). It time permits, we obtain Bernstein and Delaunay theorems for a properly immersed, horospherically convex hypersurface in $H^{n+1}$. We note that Berstein type theorem (for hypersurfaces) can be seen as Liouville type theorem (for conformal metrics).