Uniqueness Theorems for fully nonlinear conformal equations on subdomains of the sphere

TL;DR

This paper classifies solutions to fully nonlinear conformal equations on subdomains of the sphere, extending previous results to new geometries with prescribed boundary mean curvature.

Contribution

It provides new classification theorems for fully nonlinear conformal equations on spherical subdomains with specific boundary conditions, generalizing earlier work.

Findings

Classified solutions on hemispheres and geodesic balls with constant boundary mean curvature.

Extended classification results to annular domains with minimal boundary.

Generalized previous results for dimensions n ≥ 2.

Abstract

In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.