A Note on Chern-Yamabe Problem

TL;DR

This paper introduces a flow approach to the Chern-Yamabe problem, analyzes its long-term behavior, and explores the properties of an associated functional, revealing its unbounded nature and its relation to the problem.

Contribution

It presents a new flow method for the Chern-Yamabe problem and establishes the functional's properties, including monotonicity and unboundedness, in the balanced case.

Findings

Flow existence for the Chern-Yamabe problem

Monotonicity of the functional along the flow

Functional is not bounded from below

Abstract

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.