Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

TL;DR

This paper proves existence and uniqueness of conformal metrics with prescribed scalar and mean curvatures on compact manifolds with boundary, using three methods including geometric flows, under negative conformal invariants.

Contribution

It introduces a flow-based approach to solve the prescribed curvature problem, establishing existence, uniqueness, and convergence results under specific conformal invariant conditions.

Findings

Existence and uniqueness of conformal metrics with prescribed curvatures.

Short and long time existence of prescribed curvature flows.

Convergence of flows to solutions with desired curvature properties.

Abstract

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $\partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant $Q(M,\pa M)$ is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.