On a fully nonlinear version of the Min-Oo Conjecture

On a fully nonlinear version of the Min-Oo Conjecture

Jose Espinar, IMPA
Fine Hall 314

Please note:  This is an additional DGGA seminar for this date.  In this talk, we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations in subdomains of the standard $n$-sphere $\s^n$ under suitable conditions on the boundary.  This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, $D(r)$ a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb{S}^n$, and totally umbilic with mean curvature bounded bellow by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. In particular, we recover the solution by F.M. Spiegel  to the Min-Oo conjecture for locally conformally flat manifolds.  As a side product, our methods in dimension $2$ provide a new proof to a classical theorem of Toponogov. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$.  This is a joint work with E. Barbosa and M.P. Cavalcante.