Evolution equations of curvature tensors along the hyperbolic geometric flow

TL;DR

This paper derives evolution equations for curvature tensors under the hyperbolic geometric flow, revealing that finite-time singularities lead to unbounded Ricci curvature, using global tensor methods.

Contribution

It introduces global tensor form evolution equations for curvature tensors along the hyperbolic geometric flow, differing from local coordinate approaches.

Findings

Derived evolution equations for curvature tensors and connection.

Showed finite-time singularities imply unbounded Ricci curvature.

Abstract

We consider the hyperbolic geometric flow $\frac{\partial^2}{\partial t^2}g(t)=-2Ric_{g(t)}$ introduced by Kong and Liu [KL]. When the Riemannian metric evolve, then so does its curvature. Using the techniques and ideas of S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection and the curvature tensors along the hyperbolic geometric flow. The method and results are computed and written in global tensor form, different from the local normal coordinate method in [DKL1]. In addition, we further show that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.