Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a oneparameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.2010 Mathematics Subject Classification. 81T20, 35S05, 35L05, 58J40, 53C50.The detailed results are stated in Thm. 7.8 and 7.10, see also Prop. 7.6 for the arguments that allow to get two-point functions for the original Klein-Gordon equation on the full spacetime (M, g) rather than for the reduced equation (1.3) on I × Σ.Since one can get many regular states out of a given one by applying suitable Bogoliubov transformations as in [GW1], Thm. 1.2 yields in fact a large class of Hadamard states.1.3. From quantum fields to spacetime geometry. In our approach, microlocal splittings are obtained by settingis defined by formula (1.6) with b ± (t) constructed for t ∈ I as approximate solutions (i.e. modulo smoothing terms) of the operatorial equation (1.7)∂ t + ib ± + r • ∂ t − ib ± = ∂ 2 t + r∂ t + a, and satisfying some additional conditions, see Sect. 6 (in particular Thm. 6.1) for details. We note that the approximate factorization (1.7) was already used by Junker in his construction of Hadamard states [Ju1, Ju2]. h 1,sϕ ∈ S 0 KdS , resp. ∈ S −1,0 K

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“…where {F x } x∈M is a bounded family in C ∞ b (B n−1 (0, 1)). The following fact is shown in [El,Lemma 2.27].…”

Section: Bounded Tensorsmentioning

confidence: 85%

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

Gérard

Oulghazi

Wrochna

2017

Commun. Math. Phys.

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“…where the volume on ( ) is taken with respect to the natural metric on . [27,28]. In particular a hypersurface with unit normal field is controlled if, only if, it is a bounded geometry hypersurface as in Definition 2.4.…”

Section: Controlled Submanifolds Are Of Bounded Geometrymentioning

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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“…We have shown in [10] that (H, g| H ) is then a manifold of bounded geometry in its own right. See also [24,25] for a larger class of submanifolds of manifolds with bounded geometry. Definition 2.5.…”

Section: 2mentioning

confidence: 99%

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Große

Nistor

2019

Potential Anal

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We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P, C) is equivalent to the uniform regularity of the natural family (Px, Cx) of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincaré inequality of (Ammann-Große-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, wellknown theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition. 12 4. Variational boundary conditions and regularity 13 4.

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Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Cited by 14 publications

References 60 publications

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

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

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

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