Prescribed Scalar Curvature Problem under Conformal Deformation of A Riemannian Metric with Dirichlet Boundary Condition

TL;DR

This paper proves the existence of conformal metrics with prescribed scalar curvature on compact Riemannian manifolds with boundary, extending classical results to boundary cases and solving related Yamabe equations with Dirichlet conditions.

Contribution

It establishes new existence results for conformal metrics with prescribed scalar curvature on manifolds with boundary, extending Kazdan-Warner type theorems to these settings.

Findings

Existence of conformal metrics with constant scalar curvature in interior and boundary.

Extension of Kazdan-Warner's Trichotomy Theorem to manifolds with boundary.

Solution of Yamabe equations with Dirichlet boundary conditions.

Abstract

In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy Theorem" on a different type of manifolds. The key step of these problems is to obtain a positive, smooth solution of a Yamabe equation with Dirichlet boundary conditions.