Positivity of Curvature on Manifolds with Boundary

TL;DR

This paper develops a method to deform Riemannian metrics on manifolds with boundary, preserving curvature conditions and achieving totally geodesic boundaries under certain boundary data assumptions.

Contribution

It introduces a construction of metric families that interpolate between given metrics near the boundary, maintaining curvature conditions and enabling boundary geometric modifications.

Findings

Constructed metric families preserve curvature conditions.

Deformed metrics can have totally geodesic boundaries.

Method applies under specific boundary data assumptions.

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 $M$ which 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.