# Mathematics Colloquium

## Wednesday, 13 November, 2013

**SPEAKER: **Prof. Jose Espinar, Princeton and IMPA, Brazil

**TITLE**: "Geometric View of Conformal PDEs"

**ABSTRACT: **In this talk we develop a global correspondence between immersed horospherically convex hypersurfaces $phi: {rm M}^nto H^{n+1}$ and complete conformal metrics $e^{2rho}g_{S^n}$ on domains $Omega$ in the boundary $S^n$ at infinity of $H^{n+1}$ such that $rho$ is the horospherical support function and that $partial_inftyphi({rm M}^n) = partialOmega$.

We establish results on when the hyperbolic Gauss map $G: {rm M}^nto S^n$ is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to improve the understandings of elliptic problems of both Weingarten hypersurfaces in $H^{n+1}$ and complete conformal metrics on domains in $S^n$ and relations between them.

For instance, we are able to obtain an explicit correspondence between Obata's Theorem (for conformal metrics) and Alexandrov Theorem (for hypersurfaces). Moroever, 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).}