Every conformal class contains a metric of bounded geometry

Müller, Olaf; Nardmann, Marc

doi:10.1007/s00208-014-1162-z

Cited by 12 publications

(14 citation statements)

References 15 publications

(24 reference statements)

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“…2 Because of [28], this result is true without curvature and injectivity radius assumptions. The main achievement here is the second order control of the conformal factor.…”

Section: Introduction and Main Resultsmentioning

confidence: 82%

“…Suppose first that q = mp/(m − p). Since the assumptions of Theorem 7.1 are satisfied, we have the validity of (28). Reasoning as in the proof of Lemma 7.3, one get the existence of R > 0 such that for all x ∈ M \ B g R (o), ρ > 0, one has…”

Section: A Disturbed Sobolev Inequalitymentioning

confidence: 88%

“…On the other hand, volg(Bg 1 (x)) is also uniformly upper bounded on M by Bishop-Gromov theorem, since Ricg is bounded. In particular v α is uniformly lower bounded by a positive constant, and v β is uniformly upper bounded, so that the conclusion follows from (28). The case q ∈ [p, mp/(m − p)) can be treated similarly, using the standard Sobolev inequality instead of (26).…”

Section: A Disturbed Sobolev Inequalitymentioning

confidence: 96%

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Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Impera

Rimoldi

Veronelli

2019

International Mathematics Research Notices

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We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W 2,p . The result is improved for p = 2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian. ContentsProblem 1.1. Under which (geometric) assumptions on M does one have that W 2,p 0 (M ) = W 2,p (M )? Classical results on this topic can be found in [6], [24] and references therein. In the following proposition we collect the most up-to-date achievements: point (I) was shown by E. Hebey, [23, Theorem 2.8]; point (II) was proved by B. Güneysu in [18, Proposition III.18]; point (III) is due to L. Bandara, [7] (for an alternative proof see also [18, Proposition III.18]). Proposition 1.2. Let (M m , g) be a complete Riemannian manifold.

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“…2 Because of [28], this result is true without curvature and injectivity radius assumptions. The main achievement here is the second order control of the conformal factor.…”

Section: Introduction and Main Resultsmentioning

confidence: 82%

Section: A Disturbed Sobolev Inequalitymentioning

confidence: 88%

Section: A Disturbed Sobolev Inequalitymentioning

confidence: 96%

See 1 more Smart Citation

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Impera

Rimoldi

Veronelli

2019

International Mathematics Research Notices

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“…This definition extends to vector fields on general manifolds via charts and partitions of unity. 4 That is, the injectivity radius of (M, ·, · ) is positive and each iterated covariant derivative of the curvature is uniformly bounded in the metric; see [22,45] for more details. This is automatically the case if M is compact or Euclidean.…”

Section: Diffeomorphism Groups and The Modified Constantin-lax-majda mentioning

confidence: 99%

Vanishing Distance Phenomena and the Geometric Approach to SQG

Bauer

Harms

Preston

2019

Arch Rational Mech Anal

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In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin-Lax-Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance. 15 References 15

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“…An important example of manifolds with bounded geometry is that of asymptotically hyperbolic manifolds, which play an increasingly important role in geometry and physics [5,15,24,34]. It was shown in [52] that every Riemannian manifold is conformal to a manifold with bounded geometry. Moreover, manifolds of bounded geometry can be used to study boundary value problems on singular domains, see, for instance, [17,18,22,23,38,46,[53][54][55] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Ammann

Große

Nistor

2019

Mathematische Nachrichten

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Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator Δ:Hk+1false(Mfalse)∩H01false(Mfalse)→Hk−1false(Mfalse), k∈double-struckN0, invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let ∂DM⊂∂M be an open and closed subset of the boundary of M. We say that (M,∂DM) has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance dist(x,∂DM) from a point x∈M to ∂DM⊂∂M is bounded uniformly in x (and hence, in particular, ∂DM intersects all connected components of M). For manifolds (M,∂DM) with finite width, we prove a Poincaré inequality for functions vanishing on ∂DM, thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds (M,∂DM) with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces Hsfalse(Mfalse), s≥0.

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Every conformal class contains a metric of bounded geometry

Cited by 12 publications

References 15 publications

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Vanishing Distance Phenomena and the Geometric Approach to SQG

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

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