A Note on Conformal Ricci Flow

TL;DR

This paper analyzes conformal Ricci flow, proving short-term existence on various manifolds, demonstrating monotonicity of Yamabe constant, and establishing it as a gradient flow for ADM mass.

Contribution

It introduces a new approach to conformal Ricci flow using DeTurck's trick, proving existence results, and identifying its gradient flow structure for ADM mass.

Findings

Short-time existence on compact and asymptotically flat manifolds

Monotonic increase of Yamabe constant along the flow

Conformal Ricci flow as the gradient flow for ADM mass

Abstract

In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds.