Scalar and Mean Curvature Comparison on Compact Cylinder

TL;DR

This paper investigates conditions under which a compact cylinder with a Riemannian metric can be deformed to have positive scalar curvature on its boundary, generalizing Gromov and Lawson's earlier results for tori.

Contribution

It extends scalar and mean curvature comparison results to compact cylinders with boundary, under specific angle and curvature conditions, broadening previous work on tori.

Findings

Existence of a metric with positive scalar curvature on boundary under angle conditions.

Negative mean curvature must occur if boundary cannot admit PSC metric.

Generalization of Gromov-Lawson's result to higher-dimensional cylinders.

Abstract

Let $ X $ be a closed, oriented Riemannian manifold. Denote by $ (M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g) $ a compact cylinder with smooth boundary, $ \dim M \geqslant 3 $. In this article, we address the following question: If $ g $ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $ M $ and nonnegative mean curvature on $ \partial M $; and (ii) the $ g $-angle between normal vector field $ \nu_{g} $ along $ \partial M $ and $ \partial_{\xi} \in \Gamma(TI) $ being less than $ \frac{\pi}{4} $, then there exists a metric $ \tilde{g} $ on $ M $ such that $ \tilde{g} |_{X \times \lbrace 0 \rbrace} $ is a PSC metric on $ X \cong X \times \lbrace 0 \rbrace $. Equivalently, we show that if $ X $ admits no PSC metric, but $ M $ admits a PSC metric $ g $ satisfying the angle condition, then the mean curvature on $ \partial M $ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $ X = \mathbb{T}^{n} $.