Liouville Type theorem for conformal equations on the hemisphere with prescribed mean curvature

Orateur: ESPINAR Jose
Localisation: IMPA, Brésil
Type: Séminaire de géométrie
Site: Hors LAMA , IMJ P7
Salle: 2015
Date de début: 12/05/2014 - 13:45
Date de fin: 12/05/2014 - 13:45

In this talk we will deal with the following problem. Take $\rho\in C^3 (\overline{\mathbb{S}_+^n})$, here $\overline{\mathbb{S}_+^n}$ denotes the closed hemisphere, and let us denote by $\lambda(p))=(\lambda_1(p), . . . ,\lambda_n(p))$ the eigenvalues of the Schouten tensor (at $p\in\overline{\mathbb{S}_+^n}$) of $g =e^{2\rho}g_0$ and let $f$ be a curvature function which is elliptic for conformal metrics. A natural question in geometric PDE is whether given a constant $c\in\mathbb{R}$ there exists a function $\rho\in C^3 (\overline{\mathbb{S}_+^n})$ so that \begin{equation}\tag{1} \begin{cases} f(\lambda(p)=1,& \lambda(p)\in \Gamma, p\in\mathbb{S}_+^n\\ e^{-\rho}\frac{\partial\rho}{\partial \nu}=c,& \text{on }\partial\mathbb{S}_+^n \end{cases} \end{equation} where $\nu$ is the inward unit normal. That is, if there exists a conformal metric on $\mathbb{S}_+^n$ with $f (\lambda_1 . . . ,\lambda_n ) = 1$ and constant mean curvature $c$ on $\partial\mathbb{S}_+^n$ . We will show: THEOREM. Let $\rho\in C^3 (\overline{\mathbb{S}_+^n})$ be a solution to $(1)$. Then, up to a dilatation, $g = e^{2\rho} g_0$ is given by $\Phi^∗ g_0$ on $\overline{\mathbb{S}_+^n}$ , where $\Phi$ is a conformal diffeomorphism of $(\mathbb{S}^n , g_0 )$.