Eigenvalues and entropies under the harmonic-Ricci flow

TL;DR

This paper investigates the behavior of eigenvalues and entropies during the harmonic-Ricci flow, providing new proofs, monotonicity formulas, and variations for these geometric quantities.

Contribution

It offers an alternative proof for harmonic-Ricci breathers and derives new monotonicity formulas and entropy variations under the flow.

Findings

Alternative proof for harmonic-Ricci breathers.

Monotonicity formulas for Laplacian eigenvalues.

First variation formulas for harmonic-Ricci entropies.

Abstract

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eigenvalues of Laplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of the shrinker and expanding entropies of the harmonic-Ricci flow.