The exponential map at a cuspidal singularity

Grandjean, Vincent; Grieser, Daniel

doi:10.1515/crelle-2015-0020

Cited by 5 publications

(10 citation statements)

References 12 publications

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“…Although this is not so surprising to find this value linked to the combinatorics of the resolution mapping, it is of consequence for applications, especially for geodesics (as in [10]) nearby singularities. This rational number will appear somehow in the local form of the geodesic vector field.…”

Section: Local Normal Form For the Induced Metricmentioning

confidence: 88%

“…Our goal in the present paper is to focus only on the case of real analytic surface singularities from the point of view we started with in the introduction. Moreover our previous joint works [10], [11] dealing with the inner metric of singular surfaces and [9] dealing with a singular metric on a regular surface, are exemplifying the need of a description of the inner metric which is finer than Hsiang & Pati's.…”

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confidence: 99%

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Monomialization of singular metrics on real surfaces

Grandjean

2019

RACSAM

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Let B be a real analytic vector bundle of rank 2 over a smooth real analytic surface S, equipped with a real analytic fiber-metric g and such that there exists a real analytic mapping of vector bundles T S → B inducing an isomorphism outside a proper sub-variety of S. Let κ be a real analytic 2-symmetric tensor field on B. Our main result, Theorem 9.2, roughly states the following: There exists a locally finite composition of points blowings-up σ : S → S such that there exists a unique pair of real analytic singular foliations F1 and F2 on Sonly with simple singularities adapted to the exceptional divisor E and orthogonal for the (regular extension of the) pull back on S of the fiber-metric g (only semi-positive definite along E) -locally simultaneously diagonalizing the pull-back on S of the original 2-symmetric tensor field κ.When S is the resolved surface of an embedded resolution of singularities of an embedded real analytic surface singularity S0 our result thus yields a local presentation of the extension h of the pull-back on S of the inner-metric of S0 at any point of the exceptional divisor E. We furthermore recover that the pulled-back inner metric h is locally quasi-isometric to the sum of the (symmetric tensor) square of differentials of (independent) monomials in the exceptional divisor E, namely, Hsiang & Pati property is satisfied at every point of the resolved surface S.CONTENTS 10.4. Local normal form for the induced metric 32 References 34such that there exists, up to permutation, a pair of O X 2 -invertible sub-modules Θ 1 and Θ 2 of Ω 1 X 2 , such that i) Each foliation Θ i admits only simple singularities adapted to E 2 . ii) Every point a 2 of X 2 admits an open neighborhood U 2 ∋ a 2 such that if σ 2 (a 2 ) is not a singular point of X 0 , then (Dσ 2 (a 2 ))(ker Θ 1 (a 2 )) and (Dσ 2 (a 2 ))(ker Θ 2 (a 2 )) are orthogonal lines of T σ 2 (a 2 ) X 0 ;iii) The pull-back of the metric g 0 | X 0 by the resolution mapping σ 2 extends on X 2 as a real analytic semi-Riemannian metric g 2 , which writes on U 2 asOf course what is also important is point iv) of the main theorem, since we unexpectedly (that is without further blowings-up) obtain the following (Corollary 10.9 and Corollary 10.12):Corollary. Under the hypotheses of Theorem 1.2 each point a 2 in E 2 admits Hsiang & Pati coordinates, namely there exists local analytic coordinates (u, v) centered at a 2 such that i) If a 2 is a smooth point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {u = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given byfor non-negative integer numbers l ≥ k.ii) If a 2 is a corner point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {uv = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given by

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Section: Local Normal Form For the Induced Metricmentioning

confidence: 88%

mentioning

confidence: 99%

Monomialization of singular metrics on real surfaces

Grandjean

2019

RACSAM

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“…Blow-up methods have also been used in the context of dynamical systems, e.g. in celestial mechanics [46], for analyzing geodesics on singular spaces [13] or in multiple time scale analysis, see for example [8], [38], [57] and the book [37], which gives an excellent overview and many more references.…”

Section: Related Literaturementioning

confidence: 99%

“…Definition of blow-up for manifolds (possibly with corners). It can be shown (see [47], [51]) that these constructions are invariant in the following sense: for model spaces X, Y as in case (4), 13 If X is a manifold with corners then a subset Y ⊂ X is called a p-submanifold if it is everywhere locally like the models (2.10) (p is for product). Therefore, Put differently, a subset Y ⊂ X is a p-submanifold if near every q ∈ Y there are local coordinates centered at q so that Y and every face of X containing q is a coordinate subspace, i.e.…”

Section: Blow-up Ofmentioning

confidence: 99%

“…For example, when doing an iterated blow-up we may choose projective coordinates after the first blow-up, or polar coordinates, and will get the same mathematics in the end. 13 The original idea that points on the front face correspond to directions at 0 can be used directly as an invariant definition: Let M be a manifold and Unoriented blow-up has the virtue of being definable purely algebraically, so it extends to other ground fields, e.g. to complex manifolds.…”

Section: Blow-up Ofmentioning

confidence: 99%

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Scales, blow-up and quasimode constructions

Grieser¹

2017

Geometric and Computational Spectral Theory

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In this expository article we show how the concepts of manifolds with corners, blow-ups and resolutions can be used effectively for the construction of quasimodes, i.e. of approximate eigenfunctions of the Laplacian on certain families of spaces, mostly exemplified by domains Ω h ⊂ R 2 , that degenerate as h → 0. These include standard adiabatic limit families and also families that exhibit several types of scaling behavior. An introduction to manifolds with corners and resolutions, and how they relate to the ideas of (multiple) scales and matching, is included.

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Geodesics orbiting a singularity

Grieser,

Lye

2023

J. Geom.

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We study the behaviour of geodesics on a Riemannian manifold near a generalized conical or cuspidal singularity. We show that geodesics entering a small neighbourhood of the singularity either hit the singularity or approach it to a smallest distance $$\delta $$ δ and then move away from it, winding around the singularity a number of times. We study the limiting behaviour $$\delta \rightarrow 0$$ δ → 0 in the second case. In the cuspidal case the number of windings goes to infinity as $$\delta \rightarrow 0$$ δ → 0 , and we compute the precise asymptotic behaviour of this number. The asymptotics have explicitly given leading term determined by the warping factor that describes the type of cuspidal singularity. We also discuss in some detail the relation between differential and metric notions of conical and cuspidal singularities.

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The exponential map at a cuspidal singularity

Cited by 5 publications

References 12 publications

Monomialization of singular metrics on real surfaces

Monomialization of singular metrics on real surfaces

Scales, blow-up and quasimode constructions

Geodesics orbiting a singularity

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