Liouville Energy on a Topological Two Sphere
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Liouville Energy on a Topological Two Sphere XiuXiong Chen · Meijun Zhu
Received: 11 April 2013 / Accepted: 20 November 2013 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013
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 E1 energy in the study of general Kähler manifolds. Keywords Uniformization theorem · Liouville energy · Moser–Trudinger–Onofri inequality · Blowup analysis Mathematics Subject Classification (2010) 35A23 · 42B37 1 Introduction 1.1 Description of the Problem Let (M, g) be a smooth Riemann surface. For any conformal new metric g1 = eu g, the corresponding Liouville energy is defined by g1 · (Rg1 dVg1 + Rg dVg ), ln (1) Lg (g1 ) = g M where Rg and Rg1 are twice the Gaussian curvatures Kg and Kg1 with respect to metrics g and g1 . Since Rg1 = e−u (−g u + Rg ),
B
X.X. Chen ( ) Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, USA e-mail: [email protected] M. Zhu Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA
X.X. Chen, M. Zhu
the Liouville energy of metric g1 can also be represented by Lg (g1 ) = |∇g u|2 + 2Rg u dVg . M
It was showed by Chow [14] and Chen [7] that Ricci flow and Calabi flow can be viewed as the gradient flow of the Liouville energy. Therefore if M is a topological two sphere, it is important to know in the study of Ricci and Calabi flow that the Liouville energy is bounded from below for any metric with fixed volume. In fact such boundedness can be used to show the global existence of Calabi flow via integral estimates (see, for example, Chen [7]), thus one can give an alternative proof of the uniformization theorem (for the hardest case: positive curvature case) via Calabi flow.1 However, present argument in the literature seems to be a tautology: the proof of the fact that the Liouville energy on a two sphere is bounded from below relies on the uniformization theorem. In fact, the proof can be summarized as follows. Let (M, g) be a topological sphere. From the uniformization theorem we know that (M, g) is conformally equivalent to the standard sphere (S 2 , g0 ), that is, there is a ϕ(x) ∈ C 2 (S 2 ) such that g = eϕ g0 . For a given metric g1 = eu g, since g1 = eu+ϕ g0 and Rg0 = 2, we have |∇g u|2 + 2Rg u dVg Lg (g1 ) = M
= Lg0 (g1 ) − Lg0 (g) ∇g (u + ϕ)2 + 4(u + ϕ) dVg − Lg (g) = 0 0 0 S2
≥ 16π · ln
1 4π
S2
e(u+ϕ) dVg0 − Lg0 (g).
The last inequality follows from the well-known Onofri inequality on S 2 . Thus the finiteness of the volume of the metric g1 implies that the Liouville energy of such metric is bounded from below. 1.2 Historic Note
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