Boundary connected sum of Escobar manifolds

TL;DR

This paper constructs connected sum manifolds with boundary that preserve zero scalar curvature and constant mean curvature boundary, using advanced nonlocal analysis techniques, advancing the understanding of boundary geometric problems.

Contribution

It develops a method to create connected sums of manifolds with prescribed boundary and interior curvature conditions, employing new nonlocal analytical tools.

Findings

Successfully constructed boundary connected sums with desired curvature properties

Utilized novel nonlocal analysis techniques for geometric problems

Results extend to other fractional curvature problems

Abstract

Let $(X_1, \bar g_1)$ and $(X_2, \bar g_2)$ be two compact Riemannian manifolds with boundary $(M_1,g_1)$ and $(M_2,g_2)$ respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum $X=X_1 \sharp X_2$ by excising half ball near points on the boundary. The resulting metric on $X$ has zero scalar curvature and a CMC boundary. We fully exploit the nonlocal aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures.