Infinite-time incompleteness of noncompact Yamabe flow

TL;DR

This paper demonstrates that the Yamabe flow on noncompact manifolds can evolve from a complete metric to an incomplete one over infinite time, revealing a new type of geometric incompleteness.

Contribution

It proves the existence of long-term solutions to the noncompact Yamabe flow that become incomplete as time approaches infinity, a novel phenomenon in geometric analysis.

Findings

Existence of long-time solutions that become incomplete

Convergence to an incomplete Riemannian metric

Demonstration of infinite-time incompleteness

Abstract

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.