Unbounded total mean curvature with prescribed boundary metric

Abstract

Bär has shown that a fixed boundary metric and a scalar-curvature lower bound do not generally bound total mean curvature from above. We extend this unboundedness to every smooth compact connected manifold of dimension $n\ge3$ with nonempty boundary and every prescribed boundary metric, using scalar-flat metrics. While Bär's examples use warped products, our construction uses a nonzero cubic variation of total mean curvature near a flat product, followed by localization and conformal correction. We also construct arbitrarily $C^k$-small perturbations of the flat torical half-cylinder with nonnegative scalar curvature, unchanged boundary metric, an exact product end, and positive total boundary mean curvature.