A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

Abstract: We 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.

Location
Deutsche Nationalbibliothek Frankfurt am Main
Extent
Online-Ressource
Language
Englisch

Bibliographic citation
A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow ; volume:4 ; number:1 ; year:2019 ; pages:30-50 ; extent:21
Geometric flows ; 4, Heft 1 (2019), 30-50 (gesamt 21)

Creator
Gursky, Matthew J.
Streets, Jeffrey

DOI
10.1515/geofl-2019-0003
URN
urn:nbn:de:101:1-2501051745308.674598775629
Rights
Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
Last update
15.08.2025, 7:30 AM CEST

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Associated

  • Gursky, Matthew J.
  • Streets, Jeffrey

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