Abstract
AbstractWe define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.
Highlights
In this paper we de ne a formal Riemannian metric on the space of metrics in a conformal class with positive curvature
We de ne a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a di erent guise
The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex
Summary
In this paper we de ne a formal Riemannian metric on the space of metrics in a conformal class with positive (or negative) curvature. This metric is related to the Mabuchi-Semmes-Donaldson [16, 25, 33] metric of Kähler geometry, as well as the Weil-Peterson-type metric introduced by Donaldson for the space of volume forms [17]. Let (M, g ) be a compact Riemannian surface with positive Gauss curvature K > , and let [g ] denote the conformal class of g. The space conformal metrics with positive Gauss curvature. Since the Gauss curvature K > , it follows that the ( , )-form K ω is positive It is closed since we are in dimension two.
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