Positivity of Curvature On Manifolds With Boundary

Abstract Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ that agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\parti...

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Bibliographic Details
Published in:International mathematics research notices 2022-07, Vol.2022 (15), p.11401-11426
Main Author: Chow, Tsz-Kiu Aaron
Format: Article
Language:English
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Summary:Abstract Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ that agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnab071