# Min-oo conjecture for fully nonlinear conformally invariant equations

## TL;DR

This paper establishes rigidity results for super-solutions to fully nonlinear conformally invariant equations on spheres, proving that under certain boundary conditions, the manifold must be isometric to a geodesic ball, with applications to classical geometric theorems.

## Contribution

It proves new rigidity theorems for fully nonlinear conformally invariant equations without assuming concavity, extending classical geometric results to broader contexts.

## Key findings

- Rigidity of super-solutions on subdomains of the sphere.
- Manifests that manifolds with boundary conditions are isometric to geodesic balls.
- Provides a new proof of Toponogov's Theorem in dimension 2.

## Abstract

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. 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 $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. As a side product, in dimension $2$ our methods provide a new proof to Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Amp\`ere equations.