A compactness theorem for scalar-flat metrics on 3-manifolds with boundary

TL;DR

This paper proves the compactness of scalar-flat metrics with constant mean curvature boundary on 3-manifolds, using blow-up analysis of a Yamabe-type boundary value problem, advancing understanding of geometric structures in conformal geometry.

Contribution

It establishes a compactness theorem for scalar-flat metrics with boundary conditions on 3-manifolds, involving new blow-up analysis techniques for Yamabe-type equations.

Findings

Proved compactness of scalar-flat conformal metrics with boundary conditions.

Developed blow-up analysis methods for Yamabe-type boundary problems.

Enhanced understanding of geometric structures in conformal geometry.

Abstract

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary.