The Dirichlet problem for Fully Nonlinear Equations Arising from Conformal Geometry

TL;DR

This paper addresses the Dirichlet problem for certain curvature equations in conformal geometry on Riemannian manifolds, establishing existence and uniqueness of solutions through a continuity method and boundary gradient estimates.

Contribution

It introduces a novel approach to obtain boundary gradient bounds using subsolutions, enabling the proof of existence and uniqueness for the problem.

Findings

Existence and uniqueness of solutions for the curvature equations.

Development of boundary gradient estimates via subsolutions.

Application of the continuity method with a priori estimates.

Abstract

We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which is based on \emph{a priori} estimates for admissible solutions. In deriving the estimates, a crucial step is to derive a lower bound for the gradient on the boundary. This is overcome by constructing a cluster of subsolutions.