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

Seunghyeok Kim, Monica Musso

Published 2021-06-17Version 1

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 linear term of the Yamabe flow on any given Riemannian manifold $M$ of dimension $N \ge 5$, the resulting flow may blow-up in the infinite time, forming singularities each of which looks like a solution of the Yamabe problem on the unit sphere $\mathbb{S}^N$. This shows that the Yamabe flow is an equation at the borderline guaranteeing the global existence and uniformly boundness of solutions. In addition, we examine the stability of the blow-up phenomena, imposing some negativity condition on the Ricci curvature at blow-up points.