The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds

Cao, Jian

doi:10.1090/s0002-9947-1991-0991959-1

Cited by 20 publications
(23 citation statements)

References 10 publications

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“…Let (M, g) be a smooth Riemannian manifold of dimension n ≥ 5, and P g be the Paneitz operator on M . For any pointX ∈ M , it was proved in [30], together with some improvement in [7] and [16], that there exists a positive smooth function κ (with control) on M such that the conformal metricg = κ We refer such coordinates as conformal normal coordinates. Notice that detg = 1 + O(|x| N ) will be enough for our use if N is sufficiently large.…”

Section: Paneitz Operator In Conformal Normal Coordinatesmentioning
confidence: 99%

Compactness of conformal metrics with constant Q-curvature. I
Li
¹
,
Xiong
²

2019
Advances in Mathematics
27
2
18
0
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We study compactness for nonnegative solutions of the fourth order constant Q-curvature equations on smooth compact Riemannian manifolds of dimension ≥ 5. If the Q-curvature equals −1, we prove that all solutions are universally bounded. If the Q-curvature is 1, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension ≤ 9. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compactness in C 4 . Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension ≤ 7, when the Yamabe invariant is positive. We also prove that, for dimension ≥ 8, the Weyl tensor has to vanish at possible blow up points of a sequence of blowing up solutions. This implies the compactness result in dimension ≥ 8 when the Weyl tensor does not vanish anywhere. To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations.

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Section: Paneitz Operator In Conformal Normal Coordinatesmentioning
confidence: 99%

Compactness of conformal metrics with constant Q-curvature. I
Li
¹
,
Xiong
²

2019
Advances in Mathematics
27
2
18
0
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“…Here and in the sequel, the exponential map exp ξ is always intended with respect to the metric g ξ . We refer to Lee-Parker [27] for a proof of the existence of conformal normal coordinates of any finite order, see also the later proofs by Cao [11] and Günther [22] of the existence of conformal normal coordinates which are volume preserving near a given point (with no remainder term in (2.4)). For any ξ ∈ M, we let Λ ξ be the smooth positive function in M such that g ξ = Λ 2 * −2 ξ g. In both cases (locally conformally flat or not), the metric g ξ can be chosen smooth with respect to ξ and such that Λ ξ (ξ) = 1 and ∇Λ ξ (ξ) = 0.…”

Section: Scheme Of the Proofmentioning
confidence: 99%

The effect of linear perturbations on the Yamabe problem
Esposito
¹
,
Pistoia
²
,
Vétois³
2013
Math. Ann.
59
1
73
0
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In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1)Scal_g, Scal_g being the Scalar curvature of (M,g). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal

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“…L g is referred to as the conformal Laplacian, and the boundary value problem (2.1) is conformally invariant (see proposition A.2). In conformal normal coordinates (see [27,41,11]) centered at a point p, we write g(x) = exp(h(x)), where h is a smooth function taking values in the space of symmetric n × n matrices. From standard properties of conformal normal coordinates it then follows that x j h ij (x) = 0, and tr h ij (x) = O(r N ), where r = dist g (x, p) and N is arbitrarily large.…”

Section: Setting Notation and Basic Definitionsmentioning
confidence: 99%

Compactness and non-compactness for the Yamabe problem on manifolds with boundary
Disconzi
¹
,
Khuri
²

2014
Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)
21
2
7
0
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We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension n ≤ 24. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when n ≥ 25. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which will be fundamental for a study of the nonumbilic case.

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The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds

Cited by 20 publications

References 10 publications

Compactness of conformal metrics with constant Q-curvature. I

Compactness of conformal metrics with constant Q-curvature. I

The effect of linear perturbations on the Yamabe problem

Compactness and non-compactness for the Yamabe problem on manifolds with boundary

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