Blow-up phenomena for scalar-flat metrics on manifolds with boundary

TL;DR

This paper investigates the blow-up phenomena of scalar-flat metrics with constant mean curvature boundary on manifolds, providing examples in high dimensions where the set of such metrics is noncompact.

Contribution

It constructs explicit examples of noncompact sets of scalar-flat metrics with boundary in dimensions n>=25, expanding understanding of geometric analysis on manifolds with boundary.

Findings

Existence of noncompact sets of scalar-flat metrics in high dimensions

Construction of examples on the unit ball with umbilic boundary

Demonstration that these manifolds are not conformally equivalent to the unit ball

Abstract

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for which this set is noncompact. These manifolds have umbilic boundary, but they are not conformally equivalent to the unit ball.