Convergence of scalar-flat metrics on manifolds with boundary under a Yamabe-type flow

TL;DR

This paper investigates a conformal flow on compact manifolds with boundary, demonstrating convergence to scalar-flat metrics with constant boundary mean curvature in specific dimensions and under certain geometric conditions.

Contribution

It establishes convergence results for a Yamabe-type flow on manifolds with boundary, extending known results to higher dimensions and special geometric cases.

Findings

Convergence proven in dimensions up to seven.

Convergence also holds in all dimensions if the manifold is spin.

Results apply under generic geometric conditions.

Abstract

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.