Yamabe metrics on cylindrical manifolds

Akutagawa, Kazuo; Botvinnik, Boris

doi:10.1007/s000390300007

Cited by 47 publications

(97 citation statements)

References 39 publications

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“…The positivity of the operator −L n k(0) plays an important role in the main existence theorem of [3], as we now recall. We do not assert that u is bounded, nor that the new constant scalar curvature metric is conic.…”

Section: Isolated Conic Pointsmentioning

confidence: 99%

“…However, in order to accomodate some of the natural geometric applications later, we present an alternate proof of the step which uses Moser iteration to give a uniform upper bound for the minimizing sequence, by another argument related to some old ideas of Varopoulos. The proof that, in certain cases, the minimizer is strictly positive uses some ideas developed by Gursky. In the second part of this paper we expand on the theme and setting of [3] by considering in more detail the case where (M, g) is a smoothly stratified Riemannian pseudomanifold, also known as an iterated edge space. We identify the local Yamabe invariants at all point p ∈ M as higher versions of the cylindrical/conic Yamabe invariants discussed above; these are simply the global Yamabe invariants for the model spaces R k × C(Z), or (conformally) equivalently, H k+1 × Z, where Z is a compact iterated edge space with lower singular 'depth' than the original space M .…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Isolated Conic Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Yamabe problem on stratified spaces

2014

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“…If G ⊂ SO(4) is a finite subgroup acting freely on S 3 , then we let G act on S 4 ⊂ R 5 acting as rotations around the x 5 -axis. The quotient S 4 /G is then a orbifold, with two singular points, and the spherical metric g S descends to this orbifold.…”

Section: Limits Of Lebrun Metricsmentioning

confidence: 99%

Monopole metrics and the orbifold Yamabe problem

Viaclovsky¹

2010

Ann. inst. Fourier

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We consider the self-dual conformal classes on n#CP 2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkähler ALE space in dimension four. Contents 23 8. Questions 31 References 32

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“…On most noncompact manifolds, the Yamabe problem is still unsolved. However, special cases have been solved, for example, on manifolds with cylindrical ends [1].…”

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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Yamabe metrics on cylindrical manifolds

Cited by 47 publications

References 39 publications

The Yamabe problem on stratified spaces

The Yamabe problem on stratified spaces

Monopole metrics and the orbifold Yamabe problem

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

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