Local $C^0$-estimate and existence theorems for some prescribed curvature problems on complete noncompact Riemannian manifolds

TL;DR

This paper establishes local $C^0$-estimates and proves the existence of complete conformal metrics with prescribed curvature on noncompact Riemannian manifolds, advancing understanding of geometric curvature problems.

Contribution

It introduces optimal asymptotic conditions for local $C^0$-estimates and extends existence results for prescribed curvature metrics using nonlinear conformal geometry techniques.

Findings

Derived optimal local $C^0$-estimates under asymptotic conditions.

Proved existence of complete conformal metrics with prescribed curvature.

Utilized Aviles-McOwen's result and its nonlinear extension for boundary manifolds.

Abstract

In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.