Prescribing the mixed scalar curvature of a foliation

Vladimir Rovenski; Leonid Zelenko

doi:10.1007/978-3-319-04675-4__5

Prescribing the mixed scalar curvature of a foliation

Vladimir Rovenski, Leonid Zelenko

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We introduce the flow of metrics on a foliated Riemannian manifold .M; g/, whose velocity along the orthogonal (to the foliation F) distribution D is proportional to the mixed scalar curvature, Scal_mix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scal_mix (e.g., positive/negative or constant)? If the mean curvature vector of D is leaf-wise conservative, then its potential function obeys the nonlinear heat equation (Formula Presented) with a leaf-wise constant Φ and known functions β_D ≥ 0 and ψ^F_i ≥ 0. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrödinger operator) the flow of metrics admits a unique global solution, whose Scal_mix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scal_mix is negative, positive, or negative constant.

Original language	English
Title of host publication	Geometry and its Applications
Editors	Pawel Walczak, Vladimir Rovenski
Publisher	Springer New York LLC
Pages	83-123
Number of pages	41
ISBN (Electronic)	9783319046747
DOIs	https://doi.org/10.1007/978-3-319-04675-4__5
State	Published - 2014
Event	2nd International workshop Geometry and Symbolic Computation, 2013 - Haifa, Israel Duration: 15 May 2013 → 18 May 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	72
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	2nd International workshop Geometry and Symbolic Computation, 2013
Country/Territory	Israel
City	Haifa
Period	15/05/13 → 18/05/13

Bibliographical note

Publisher Copyright:
© Springer International Publishing Switzerland 2014.

Keywords

Conformal
Foliation
Mean curvature vector
Mixed scalar curvature
Parabolic PDE
Riemannian metric
Schrödinger operator
Twisted product

ASJC Scopus subject areas

General Mathematics

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10.1007/978-3-319-04675-4__5

Cite this

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title = "Prescribing the mixed scalar curvature of a foliation",

abstract = "We introduce the flow of metrics on a foliated Riemannian manifold .M; g/, whose velocity along the orthogonal (to the foliation F) distribution D is proportional to the mixed scalar curvature, Scalmix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scalmix (e.g., positive/negative or constant)? If the mean curvature vector of D is leaf-wise conservative, then its potential function obeys the nonlinear heat equation (Formula Presented) with a leaf-wise constant Φ and known functions βD ≥ 0 and ψFi ≥ 0. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schr{\"o}dinger operator) the flow of metrics admits a unique global solution, whose Scalmix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scalmix is negative, positive, or negative constant.",

keywords = "Conformal, Foliation, Mean curvature vector, Mixed scalar curvature, Parabolic PDE, Riemannian metric, Schr{\"o}dinger operator, Twisted product",

author = "Vladimir Rovenski and Leonid Zelenko",

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year = "2014",

doi = "10.1007/978-3-319-04675-4__5",

language = "English",

series = "Springer Proceedings in Mathematics and Statistics",

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pages = "83--123",

editor = "Pawel Walczak and Vladimir Rovenski",

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Rovenski, V & Zelenko, L 2014, Prescribing the mixed scalar curvature of a foliation. in P Walczak & V Rovenski (eds), Geometry and its Applications. Springer Proceedings in Mathematics and Statistics, vol. 72, Springer New York LLC, pp. 83-123, 2nd International workshop Geometry and Symbolic Computation, 2013, Haifa, Israel, 15/05/13. https://doi.org/10.1007/978-3-319-04675-4__5

Prescribing the mixed scalar curvature of a foliation. / Rovenski, Vladimir; Zelenko, Leonid.
Geometry and its Applications. ed. / Pawel Walczak; Vladimir Rovenski. Springer New York LLC, 2014. p. 83-123 (Springer Proceedings in Mathematics and Statistics; Vol. 72).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N1 - Publisher Copyright: © Springer International Publishing Switzerland 2014.

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N2 - We introduce the flow of metrics on a foliated Riemannian manifold .M; g/, whose velocity along the orthogonal (to the foliation F) distribution D is proportional to the mixed scalar curvature, Scalmix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scalmix (e.g., positive/negative or constant)? If the mean curvature vector of D is leaf-wise conservative, then its potential function obeys the nonlinear heat equation (Formula Presented) with a leaf-wise constant Φ and known functions βD ≥ 0 and ψFi ≥ 0. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrödinger operator) the flow of metrics admits a unique global solution, whose Scalmix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scalmix is negative, positive, or negative constant.

AB - We introduce the flow of metrics on a foliated Riemannian manifold .M; g/, whose velocity along the orthogonal (to the foliation F) distribution D is proportional to the mixed scalar curvature, Scalmix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scalmix (e.g., positive/negative or constant)? If the mean curvature vector of D is leaf-wise conservative, then its potential function obeys the nonlinear heat equation (Formula Presented) with a leaf-wise constant Φ and known functions βD ≥ 0 and ψFi ≥ 0. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrödinger operator) the flow of metrics admits a unique global solution, whose Scalmix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scalmix is negative, positive, or negative constant.

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KW - Mean curvature vector

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KW - Parabolic PDE

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KW - Schrödinger operator

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ER -

Prescribing the mixed scalar curvature of a foliation

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Keywords

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