Liouville Energy on a Topological Two Sphere

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 Kähler manifolds.1 In a recent note, Chen, Lu and Tian [11] clarify that the Ricci flow did yield another proof of the uniformization theorem for spheres even without using current integral estimates. Nevertheless our result not only answer a conjecture of Chen [8], but also has broad applications in the study of Kähler manifolds (see the historic note).

A generalized Trudinger–Moser inequality on a compact Riemannian surface

“…In view of (56), if one has (8) below, then we have J ψ,h (φ ǫ ) < inf u∈W 1,2 (Σ,g) J ψ,h (u) for sufficiently small ǫ > 0. Then Theorem 2 tells us that no blowup happens, so J ψ,h achieves its infimum at some function u ∈ W 1,2 (Σ, g).…”

“…The Liouville energy of metric g = e u g with respect to metric g is represented as L g ( g) = Σ |∇ g u| 2 dv g + 4 Σ K g udv g . When (Σ, g) is a topological two sphere with volume 4π and bounded curvature K g , Chen-Zhu [8] proved that L g ( g) is bounded from below in W 1,2 (Σ, g). Their proof is analytic, does not rely on the uniformization theorem and the Onofri inequality.…”

Convergence of the Calabi Flow on Toric Varieties and Related Kähler Manifolds

“…When the complex dimension of X is one, Conjecture (1.2) is proved by Chen [8] and Chen-Zhu [12]. When X is a toric surface, Conjecture (1.2) is proved in [23] if (X, P ) is analytic relative K-stable.…”

A Trudinger–Moser Inequality on a Compact Riemannian Surface Involving Gaussian Curvature

“…The first and the most important result in this paper can be stated as follows: 2 Theorem 1. Let (Σ, g) be a compact Riemannian surface without boundary, K g be its Gaussian curvature, and K g , λ g (Σ) be defined as in (8), (9) respectively. Suppose that the Euler characteristic χ(Σ) 0.…”

“…Furthermore, the following theorem reveals the relation between the Trudinger-Moser inequality and the topology of Σ. Theorem 2. Let (Σ, g) be a compact Riemannian surface without boundary, K g be defined as in (8). Then the Trudinger-Moser inequality…”