A compactness theorem for scalar-flat metrics on manifolds with boundary

TL;DR

This paper proves a compactness theorem for scalar-flat metrics conformal to a given metric on manifolds with boundary, under generic conditions, in dimensions seven and higher.

Contribution

It establishes the compactness of the set of scalar-flat conformal metrics with constant mean curvature boundary in higher dimensions, assuming a generic boundary condition.

Findings

Set of scalar-flat metrics is compact in dimensions ≥7.

Compactness holds under the condition that the trace-free 2nd fundamental form is nonzero everywhere.

Results apply to manifolds with boundary in conformal geometry.

Abstract

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.