Vipul: 3 revisions

2008-05-18T20:12:50Z

3 revisions

← Older revision	Revision as of 20:12, 18 May 2008
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Vipul at 15:25, 2 September 2007

2007-09-02T15:25:38Z

← Older revision		Revision as of 15:25, 2 September 2007
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	==Statement==		==Statement==

	~~Any~~ [[surface]] (viz, two-dimensional [[differential manifold]]) ~~admits~~ a [[Riemannian metric]] of constant [[Gaussian curvature]] ~~(which, for a surface,~~ is ~~the same as a~~ [[~~constant-~~curvature ~~metric~~]]~~). More strongly~~, ~~given a surface, and a~~ [[~~conformal class of Riemannian metrics~~]] ~~on that surface, there exists a~~ [[~~constant-~~curvature ~~metric~~]] ~~in that conformal class~~.		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]]}}

Vipul at 08:56, 2 September 2007

2007-09-02T08:56:44Z

← Older revision		Revision as of 08:56, 2 September 2007
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	===Geometrization conjecture===		===Geometrization conjecture===

			{{further\|[[Geometrization conjecture]]}}

Vipul at 08:56, 2 September 2007

2007-09-02T08:56:30Z

New page

==Statement==

Any [[surface]] (viz, two-dimensional [[differential manifold]]) admits a [[Riemannian metric]] of constant [[Gaussian curvature]] (which, for a surface, is the same as a [[constant-curvature metric]]). More strongly, given a surface, and a [[conformal class of Riemannian metrics]] on that surface, there exists a [[constant-curvature metric]] in that conformal class.

==Relation with other results==

===Geometrization conjecture===

Uniformization theorem - Revision history

Vipul: 3 revisions

Vipul at 15:25, 2 September 2007

Vipul at 08:56, 2 September 2007

Vipul at 08:56, 2 September 2007