On the partial uniform ellipticity and complete conformal metrics with prescribed curvature functions on manifolds with boundary

TL;DR

This paper investigates the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary, revealing algebraic structures of nonlinear equations and addressing topological obstructions.

Contribution

It introduces a novel algebraic framework for fully nonlinear equations related to conformal metrics with prescribed curvature, including analysis of topological obstructions.

Findings

Established algebraic structures for nonlinear curvature equations

Identified topological obstructions to metric existence

Confirmed key assumptions for Hessian and Weingarten equations

Abstract

We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.