A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

TL;DR

This paper proves the compactness of scalar-flat metrics with umbilic boundary on low-dimensional manifolds (n=6,7,8) under certain conditions, contributing to the understanding of geometric structures in conformal geometry.

Contribution

It establishes a compactness result for scalar-flat metrics with umbilic boundary in low dimensions when the Weyl tensor does not vanish on the boundary.

Findings

Scalar-flat metrics form a compact set in low dimensions n=6,7,8.

Non-vanishing Weyl tensor on the boundary is crucial for compactness.

Results extend understanding of conformal metrics with boundary conditions.

Abstract

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n=6,7,8, provided that the Weyl tensor is always not vanishing on the boundary.