Local geometry of singular real analytic surfaces

Grieser, Daniel

doi:10.1090/s0002-9947-02-03168-9

Cited by 7 publications

(5 citation statements)

References 26 publications

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“…If q ∈ supp(τ i ) then q ∈ π −1 Y i (U p i ) and using (5) we get φ(q) = (u, [r, y]) with u ∈ U p i and [r, y] ∈ sing(C(L Y )). We have (τ i ψ Up i ,n ) • φ −1 = (τ i • φ −1 )α Up i ,n where α Up i ,n is defined in (18). By construction (τ i • φ −1 )α Up i ,n is null on a neighborhood (which depends on n) of (u, [r, y]) because…”

Section: Stratified Pseudomanifolds With Iterated Edge Metricsmentioning

confidence: 99%

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q-parabolicity of stratified pseudomanifolds and other singular spaces

Bei

Güneysu

2016

Ann Glob Anal Geom

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The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with aĉ-iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compact complex Hermitian spaces, is investigated. 0 q-PARABOLICITY OF STRATIFIED PSEUDOMANIFOLDS AND OTHER SINGULAR SPACES 1 form M/G with M a compact manifold and G a compact Lie group acting isometrically), this result entails that these spaces are automatically 2-parabolic and thus stochastically complete. The importance of this class of metrics, as we will explain more precisely later, lies in its deep connection with the topology of X. Besides to stratified pseudomanifolds, in this paper we investigate also the q-parabolicity of other classes of singular spaces such as compact Hermitian complex spaces, real algebraic varieties and almost complex manifolds endowed with a compatible and degenerate metric whose degeneration locus is a union of closed submanifolds with codimension ≥ 2. Let us point out that all of the above examples provide smooth Riemannian manifolds which are geodesically incomplete, so that one cannot use Grigoryan's well-known parabolicity and stochastic completeness criteria (cf. Theorem 11.8 and Theorem 11.14 in [21]) which require geodesic completeness and volume control. Our approach is more in the spirit of [22]. This paper is organized as follows: In the second section we recall the main definitions and some important properties concerning parabolicity of Riemannian manifolds. The third section, which contains the main result of this paper, deals with compact stratified pseudomanifolds. The forth section is divided in three parts. The first one contains some technical statements that will be extensively used through the rest of the paper. The second part deals with almost complex manifolds endowed with a compatible and degenerate metric whose degeneration locus is a union of closed submanifolds with codimension ≥ 2. Finally the third part tackles the case of compact Hermitian complex spaces and real algebraic varieties .

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Section: Stratified Pseudomanifolds With Iterated Edge Metricsmentioning

confidence: 99%

“…) holds. Define now a sequence on U p × C(L Y ) as(18) α Up,n := γ Up,n β L Y ,n .Weclearly have lim n→∞ α Up,n (x) = 1 for every x ∈ U p × C(L Y ). Over U p × reg(C(L Y )), for d(α Up,n ), we have dα Up,n = γ Up,n dβ Up,n + β Up,n dγ Up,n…”

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confidence: 99%

q-parabolicity of stratified pseudomanifolds and other singular spaces

Bei

Güneysu

2016

Ann Glob Anal Geom

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“…If the real codimension of Σ is greater than 1, then Σ is almost polar ( [20], [35], and [21]), • there exists M such that ∂ C M is almost polar but M does not have negligible boundary [11] (see also [3] and [24]), • if M is Kähler and ∂ C M is almost polar, then M has negligible boundary [10], [26]. The last item follows from an estimate of the cut-off function's gradient near the Cauchy boundary, which could be proved by applying the Kähler identity.…”

Section: Lemma 5 ([22])mentioning

confidence: 99%

Vanishing and conservativeness of harmonic forms of a non-compact CR manifold

Masamune¹

2008

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…For example, the description of the asymptotic behavior of the exponential map may be used to deduce the asymptotic behavior of the volume of balls centered at the singularity, as the radius tends to zero. In the case where p is a smooth point, the coefficients in this expansion are related to the curvature at p. In [Gri2] such an expansion was derived for real analytic isolated surface singularities; however, the balls were defined extrinsically, i.e. with distance defined as distance in the ambient space R n .…”

Section: Introductionmentioning

confidence: 99%

The exponential map at a cuspidal singularity

Grandjean¹,

Grieser

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.

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Local geometry of singular real analytic surfaces

Cited by 7 publications

References 26 publications

q-parabolicity of stratified pseudomanifolds and other singular spaces

q-parabolicity of stratified pseudomanifolds and other singular spaces

Vanishing and conservativeness of harmonic forms of a non-compact CR manifold

The exponential map at a cuspidal singularity

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