Concentration on minimal submanifolds for a Yamabe type problem

TL;DR

This paper constructs solutions to a Yamabe type problem on higher-dimensional manifolds that concentrate around minimal submanifolds under specific geometric conditions, linking geometric analysis with PDEs.

Contribution

It introduces a novel method to produce solutions concentrating on minimal submanifolds for Yamabe problems with near-critical nonlinearities, under curvature conditions.

Findings

Solutions concentrate around nondegenerate minimal submanifolds.

A geometric condition involving sectional curvatures is necessary.

Connection established with PDEs on submanifolds with singular terms.

Abstract

We construct solutions to a Yamabe type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a non degenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.