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

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.