Gluing Scalar-Flat Manifolds with Vanishing Mean Curvature on the Boundary

TL;DR

This paper develops a gluing technique for scalar-flat manifolds with boundary, constructing new metrics with controlled mean curvature properties, advancing the understanding of boundary value problems in geometric analysis.

Contribution

It introduces a gluing theorem for scalar-flat manifolds with boundary, enabling the construction of new solutions with prescribed boundary mean curvature.

Findings

Constructed scalar-flat metrics with small, constant boundary mean curvature.

Achieved vanishing boundary mean curvature under additional geometric conditions.

Extended the class of manifolds where the Yamabe problem with boundary can be solved.

Abstract

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we produce the generalized connected sum along $K$. On this third manifold we produce a family of scalar-flat metrics with small, constant mean curvature on the boundary which are close to the original metrics in the $C^2$ sense. Under extra geometric conditions on the original manifolds, we can arrange for this family to also have vanishing mean curvature on the boundary.