Generalized connected sum construction for constant scalar curvature metrics

TL;DR

This paper develops a method to construct constant scalar curvature metrics on connected sums of manifolds along a common submanifold, extending the Yamabe problem to more complex geometric configurations.

Contribution

It introduces a generalized connected sum construction for constant scalar curvature metrics along higher codimension submanifolds.

Findings

Successfully constructs solutions to the Yamabe equation on generalized connected sums.

Extends existing methods to higher codimension cases.

Provides a new approach for conformal geometry problems.

Abstract

We consider the problem of constructing solutions to the Yamabe equation (i.e. conformal constant scalar curvature metrics) on the generalized connected sum M = (M_1) #_K (M_2) of two compact Riemannian manifolds (M_1,g_1) and (M_2,g_2) along a common (isometrically embedded) submanifold (K,g_K) of codimension greater or equal than 3.