Uniformization theorem: Difference between revisions
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==Statement== | ==Statement== | ||
Given a [[surface]] (viz, two-dimensional [[differential manifold]]) and a [[conformal class of Riemannian metrics]] on that surface, there exists a [[constant-curvature metric]] in that conformal class. Note that for surfaces, having constant [[Gaussian curvature]] is equivalent to having constant [[sectional curvature]], constant [[Ricci curvature]] or constant [[scalar curvature]]. | |||
==Relation with other results== | ==Proof== | ||
One proof of the uniformization conjecture relies on the [[volume-normalized Ricci flow]] in two dimensions, which is the same as the [[Yamabe flow]]. The idea is to show that starting with any [[Riemannian metric]], we can, using the Yamabe flow, evolve it without changing the conformal class, to attain a constant-curvature metric in the limit. | |||
==Relation with other results/conjectures== | |||
===Theorems on Ricci flows=== | |||
[[Hamilton's theorem on Ricci flows]] along with [[Chow's theorem]] provide the Ricci flow-route to proving the uniformization theorem. | |||
===Yamabe conjecture=== | |||
{{further|[[Yamabe conjecture]]}} | |||
The Yamabe conjecture is an attempt to generalize the uniformization theorem to higher dimensions. Here, we use [[scalar curvature]] to generalize the [[Gaussian curvature]] for surfaces. | |||
===Geometrization conjecture=== | ===Geometrization conjecture=== | ||
{{further|[[Geometrization conjecture]]}} | {{further|[[Geometrization conjecture]]}} | ||
Latest revision as of 20:12, 18 May 2008
Statement
Given a surface (viz, two-dimensional differential manifold) and a conformal class of Riemannian metrics on that surface, there exists a constant-curvature metric in that conformal class. Note that for surfaces, having constant Gaussian curvature is equivalent to having constant sectional curvature, constant Ricci curvature or constant scalar curvature.
Proof
One proof of the uniformization conjecture relies on the volume-normalized Ricci flow in two dimensions, which is the same as the Yamabe flow. The idea is to show that starting with any Riemannian metric, we can, using the Yamabe flow, evolve it without changing the conformal class, to attain a constant-curvature metric in the limit.
Relation with other results/conjectures
Theorems on Ricci flows
Hamilton's theorem on Ricci flows along with Chow's theorem provide the Ricci flow-route to proving the uniformization theorem.
Yamabe conjecture
Further information: Yamabe conjecture
The Yamabe conjecture is an attempt to generalize the uniformization theorem to higher dimensions. Here, we use scalar curvature to generalize the Gaussian curvature for surfaces.
Geometrization conjecture
Further information: Geometrization conjecture