Prescribed curvature flow on surfaces

Ho, Pak Tung

doi:10.1512/iumj.2011.60.4413

Cited by 18 publications

(18 citation statements)

References 7 publications

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“…More precisely, given a function f on a compact n-dimensional Riemannian manifold (M, g 0 ) without boundary, can we find a conformal metric g such that R g = f ? This prescribed scalar curvature problem has been studied extensively, see [26,28,29] and references therein. Especially, the problem has attracted a lot of attention when (M, g 0 ) is the n-dimensional standard sphere S n , which is the so-called Kazdan-Warner's problem for n ≥ 3 and Nirenberg's problem for n = 2.…”

Section: Introductionmentioning

confidence: 99%

Conformal curvature flows on compact manifold of negative Yamabe constant

Chen¹,

Ho²

2018

Indiana Univ. Math. J.

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We study some conformal curvature flows related to prescribed curvature problems on a smooth compact Riemannian manifold (M, g 0) with or without boundary, which is of negative (generalized) Yamabe constant, including scalar curvature flow and conformal mean curvature flow. Using such flows, we show that there exists a unique conformal metric of g 0 such that its scalar curvature in the interior or mean curvature curvature on the boundary is equal to any prescribed negative smooth function, which partially recovers the results of Kazdan-Warner, Aubin and Escobar. We also study the soliton to some Yamabe-type flow on a compact manifold with smooth boundary.

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Section: Introductionmentioning

confidence: 99%

Conformal curvature flows on compact manifold of negative Yamabe constant

Chen¹,

Ho²

2018

Indiana Univ. Math. J.

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“…More precisely, given a function f on a compact n-dimensional Riemannian manifold (M, g 0 ), can we find a metric g conformal to g 0 such that R g = f ? This prescribing scalar curvature problem has been studied extensively, even for the case when M is a surface, see [26,33,34]. Especially, the problem has attracted a lot of attention when (M, g 0 ) is the n-dimensional standard sphere S n , which is the so-called Nirenberg's problem.…”

Section: Introductionmentioning

confidence: 99%

The Webster scalar curvature flow on CR sphere. Part I

2015

Advances in Mathematics

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This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper.Comment: To appear in Advances in Mathematic

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“…In Riemannian geometry, the problem of finding a conformal metric on a compact Riemannian manifold with a prescribed scalar curvature has been investigated extensively (cf. [KW1,KW2,KW3,Ou1,Ou2,Ta,Ra,Ho1,CX] and the references therein). Its special case that the candidate scalar curvature function is constant is the well-known Yamabe problem, which was settled down by a series of works due to Yamabe, Trudinger, Aubin, and Schoen (cf.…”

Section: Introductionmentioning

confidence: 98%

Prescribed Webster scalar curvatures on compact pseudo-Hermitian manifolds

Yan¹,

Ren²,

Yu³

2021

Preprint

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In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of selfadjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

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Prescribed curvature flow on surfaces

Cited by 18 publications

References 7 publications

Conformal curvature flows on compact manifold of negative Yamabe constant

Conformal curvature flows on compact manifold of negative Yamabe constant

The Webster scalar curvature flow on CR sphere. Part I

Prescribed Webster scalar curvatures on compact pseudo-Hermitian manifolds

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