Towering phenomena for the Yamabe equation on symmetric manifolds

TL;DR

This paper studies the Yamabe equation on symmetric manifolds, showing that for small perturbations, solutions form towering blow-up patterns centered at a symmetric point with non-zero Weyl tensor.

Contribution

It constructs symmetric solutions exhibiting multiple bubble blow-ups at a point on symmetric manifolds, revealing new blow-up phenomena for the perturbed Yamabe problem.

Findings

Existence of solutions with multiple bubbles for small perturbations.

Solutions concentrate at a symmetric point with non-zero Weyl tensor.

Demonstration of towering blow-up behavior in the Yamabe problem.

Abstract

Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $\xi_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_\epsilon)\qquad-\mathcal L_g u+\epsilon u=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $\epsilon_k>0$ such that for all $\epsilon\in (0, \epsilon_k)$ the problem $(P_\epsilon)$ has a symmetric solution $u_\epsilon ,$ which looks like the superposition of $k$ positive bubbles centered at the point $\xi_0$ as $\epsilon\to 0$. In particular, $\xi_0$ is a {\em towering} blow-up point.