Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

TL;DR

This paper demonstrates that small smooth perturbations to the Yamabe flow on higher-dimensional manifolds can cause solutions to blow up infinitely often over time, using explicit blow-up profiles and analyzing stability conditions.

Contribution

It constructs explicit infinite-time blow-up solutions for perturbed Yamabe flows on manifolds of dimension five or higher, extending understanding of flow singularities.

Findings

Existence of infinite-time blow-up solutions under small perturbations

Construction of blow-up profiles using solutions on the sphere

Stability analysis under Ricci curvature negativity

Abstract

Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension $N \ge 3$ is known to exist for all time $t$ and converges to a solution to the Yamabe problem as $t \to \infty$. We prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the Yamabe flow on any given Riemannian manifold $M$ of dimension $N \ge 5$, the resulting flow may blow up at multiple points on $M$ in the infinite time. Our proof is constructive, and indeed we construct such a flow by using solutions of the Yamabe problem on the unit sphere $\mathbb{S}^N$ as blow-up profiles. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.