Eldan's Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Bo'az Klartag

doi:10.1007/s12220-017-9894-0

Eldan's Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Bo'az Klartag

Weizmann Institute of Science, Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Let f: C ⁿ→ C ^k be a holomorphic function and set Z= f ^{- 1}(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γ _n is the standard Gaussian measure in C ⁿ, and E⊆ C ⁿ is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

Original language	English
Pages (from-to)	2008-2027
Number of pages	20
Journal	Journal of Geometric Analysis
Volume	28
Issue number	3
Early online date	4 Jul 2017
DOIs	https://doi.org/10.1007/s12220-017-9894-0
State	Published - 1 Jul 2018

Keywords

Complex varieties
Gaussian measure
Stochastic processes
Tubular neighborhoods

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1007/s12220-017-9894-0

http://arxiv.org/pdf/1702.02315License: Other

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title = "Eldan's Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets",

abstract = "Let f: C n→ C k be a holomorphic function and set Z= f - 1(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γ n is the standard Gaussian measure in C n, and E⊆ C n is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin. ",

keywords = "Complex varieties, Gaussian measure, Stochastic processes, Tubular neighborhoods",

author = "Bo'az Klartag",

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N2 - Let f: C n→ C k be a holomorphic function and set Z= f - 1(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γ n is the standard Gaussian measure in C n, and E⊆ C n is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

AB - Let f: C n→ C k be a holomorphic function and set Z= f - 1(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γ n is the standard Gaussian measure in C n, and E⊆ C n is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

KW - Complex varieties

KW - Gaussian measure

KW - Stochastic processes

KW - Tubular neighborhoods

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U2 - 10.1007/s12220-017-9894-0

DO - 10.1007/s12220-017-9894-0

M3 - مقالة

SN - 1050-6926

VL - 28

SP - 2008

EP - 2027

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 3

ER -

Eldan's Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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