Instantaneously complete Yamabe flow on hyperbolic space

TL;DR

This paper proves the global existence and uniqueness of instantaneously complete Yamabe flows on hyperbolic space for initial metrics with certain bounds, without requiring initial completeness or Ricci curvature bounds.

Contribution

It establishes the existence and uniqueness of Yamabe flows under broader initial conditions on hyperbolic space, including non-complete metrics.

Findings

Global existence of Yamabe flows on hyperbolic space

Uniqueness for rotationally symmetric initial data

No need for initial completeness or Ricci bounds

Abstract

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.