A class of fully nonlinear equations arising in conformal geometry

TL;DR

This paper studies a new class of fully nonlinear equations on smooth Riemannian manifolds, extending the $\sigma_k$ Yamabe equation, and provides key estimates and existence results for their solutions.

Contribution

It introduces a novel class of nonlinear equations in conformal geometry and proves fundamental estimates and existence theorems for solutions.

Findings

Established local gradient estimates

Proved second derivative bounds

Demonstrated existence of solutions

Abstract

In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for solutions to these equations and establish an existence result associated to them.