Liouville energy on a topological two sphere

TL;DR

This paper provides an analytic proof that the Liouville energy on a topological two sphere is bounded below, avoiding reliance on classical theorems, which aids in understanding energy bounds in Kähler geometry.

Contribution

It offers a novel analytic proof of the lower boundedness of Liouville energy without using the uniformization theorem or Onofri inequality.

Findings

Liouville energy on a topological two sphere is bounded from below.

The proof does not depend on the uniformization theorem or Onofri inequality.

Insights applicable to energy bounds in Kähler geometry.

Abstract

In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E_1 energy in the study of general K\"ahler manifolds.