Conformal deformation of a Riemannian metric to constant scalar curvature

Schoen, Richard

doi:10.4310/jdg/1214439291

Cited by 1,021 publications

(829 citation statements)

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“…An elegant proof of the positive mass theorem by Witten (see [65]) uses Sobolev embeddings on such manifolds. The positive mass theorem provides the final step in the proof of the Yamabe conjecture on compact manifolds [69]: any conformal class on a compact manifold M admits a metric with constant scalar curvature. In order to prove the conjecture in the locally conformally flat case, one replaces the metric g on M by a scalar-flat metric u · g on M \ {p}, where u is a function u(x) → ∞ for x → p, and a neighborhood of p provides the asymptotically Euclidean end, and one applies the positive mass conjecture to this.…”

Section: We Define ϕ : [−1 +1] → R ϕ(T) = Log(t +1)−log(1−t) ϕ(±1)mentioning

confidence: 99%

On the geometry of Riemannian manifolds with a Lie structure at infinity

Ammann

Lauter

Nistor

2004

International Journal of Mathematics and Mathematical Sciences

214

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We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.2000 Mathematics Subject Classification: 53C21, 53C27, 58J40.1. Introduction. Geometric differential operators on complete noncompact Riemannian manifolds were extensively studied due to their applications to physics, geometry, number theory, and numerical analysis. Still, their properties are not as well understood as those of differential operators on compact manifolds, one of the main reasons being that differential operators on noncompact manifolds do not enjoy some of the most useful properties enjoyed by their counterparts on compact manifolds.For example, elliptic operators on noncompact manifolds are not Fredholm in general. (We use the term "elliptic" in the sense that the principal symbol is invertible outside the zero section.) Also, one does not have a completely satisfactory pseudodifferential calculus on an arbitrary complete noncompact Riemannian manifold, which might allow us to decide whether a given geometric differential operator is bounded, Fredholm, or compact (see however [2] and the references therein).However, if one restricts oneself to certain classes of complete noncompact Riemannian manifolds, one has a chance to obtain more precise results on the analysis of the geometric differential operators on those spaces. This paper is the first in a series of papers devoted to the study of such a class of Riemannian manifolds, the class of Riemannian manifolds with a "Lie structure at infinity" (see Definition 3.1). We stress here that few results on the geometry of these manifolds have a parallel in the literature, although there are a fair number of papers devoted to the analysis on particular classes of such manifolds [12,13,15,16,17,38,39,55,57,64,67,70,74,75,76,78] A manifold M 0 with a Lie structure at infinity has, by definition, a natural compactification to a manifold with corners M = M 0 ∪∂M such that the tangent bundle T M 0 → M 0

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Section: We Define ϕ : [−1 +1] → R ϕ(T) = Log(t +1)−log(1−t) ϕ(±1)mentioning

confidence: 99%

On the geometry of Riemannian manifolds with a Lie structure at infinity

Ammann

Lauter

Nistor

2004

International Journal of Mathematics and Mathematical Sciences

214

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“…The geometric meaning of this functional is that if u is a minimizer, or indeed any critical point, then the conformally related metric g = u 4 n−2 g has constant scalar curvature on any open set where u > 0. We refer to the well-known survey paper by Lee and Parker [16], as well as [27], [28], for all details on the complete existence theory in the setting of smooth compact manifolds.…”

Section: Introductionmentioning

confidence: 99%

The Yamabe problem on stratified spaces

2014

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We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik [3] on the Yamabe problem on spaces with isolated conic singularities.

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“…In [2], Aubin improved Trudinger's result, using variational methods and Weyl's tensor characteristics. In 1984, Schoen [66] solved the remaining cases using variational methods and the positive mass Theorem. We have also to point out the work of Lee and Parker in [61], which is a detailed discussion on the Yamabe problem unifying the work of Aubin [2] with that of Schoen [66].…”

Section: The Yamabe Problemmentioning

confidence: 99%

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Gamara

2017

Arab. J. Math.

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Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.Mathematics Subject Classification 58E05 · 57R70 · 53C21 · 53C17

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Conformal deformation of a Riemannian metric to constant scalar curvature

Cited by 1,021 publications

References 5 publications

On the geometry of Riemannian manifolds with a Lie structure at infinity

On the geometry of Riemannian manifolds with a Lie structure at infinity

The Yamabe problem on stratified spaces

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

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