Scalar and Mean Curvature Comparison on Compact Cylinder

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} $.