Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

Robert L. Jerrard; Cy Maor

doi:https://doi.org/10.1007/s10455-019-09670-z

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

Robert L. Jerrard, Cy Maor

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

Abstract

We study the geodesic distance induced by right-invariant metrics on the group Diff _c(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms W^s ^, ⁿ ^/ ^s, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by W^s ^, ^p vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.

Original language	English
Pages (from-to)	351-360
Number of pages	10
Journal	Annals of Global Analysis and Geometry
Volume	56
Issue number	2
DOIs	https://doi.org/10.1007/s10455-019-09670-z
State	Published - 1 Sep 2019
Externally published	Yes

Keywords

Diffeomorphism group
Fractional Sobolev spaces
Infinite-dimensional geometry
Vanishing geodesic distance

All Science Journal Classification (ASJC) codes

Analysis
Political Science and International Relations
Geometry and Topology

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https://doi.org/10.1007/s10455-019-09670-z

Cite this

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title = "Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents",

abstract = "We study the geodesic distance induced by right-invariant metrics on the group Diff c(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms Ws , n / s, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by Ws , p vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.",

keywords = "Diffeomorphism group, Fractional Sobolev spaces, Infinite-dimensional geometry, Vanishing geodesic distance",

author = "Jerrard, {Robert L.} and Cy Maor",

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doi = "https://doi.org/10.1007/s10455-019-09670-z",

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T1 - Geodesic distance for right-invariant metrics on diffeomorphism groups

T2 - critical Sobolev exponents

AU - Jerrard, Robert L.

AU - Maor, Cy

PY - 2019/9/1

Y1 - 2019/9/1

N2 - We study the geodesic distance induced by right-invariant metrics on the group Diff c(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms Ws , n / s, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by Ws , p vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.

AB - We study the geodesic distance induced by right-invariant metrics on the group Diff c(M) of compactly supported diffeomorphisms of a manifold M and show that it vanishes for the critical Sobolev norms Ws , n / s, where n is the dimension of M and s∈ (0 , 1). This completes the proof that the geodesic distance induced by Ws , p vanishes if sp≤ n and s< 1 , and is positive otherwise. The proof is achieved by combining the techniques of two recent papers—(Jerrard and Maor in Ann Glob Anal Geom 55(4):631–656, 2019) by the authors, which treated the subcritical case, and Bauer et al. (Vanishing distance phenomena and the geometric approach to SQG, 2018. arXiv:1805.04401), which treated the critical one-dimensional case.

KW - Diffeomorphism group

KW - Fractional Sobolev spaces

KW - Infinite-dimensional geometry

KW - Vanishing geodesic distance

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DO - https://doi.org/10.1007/s10455-019-09670-z

M3 - مقالة

SN - 0232-704X

VL - 56

SP - 351

EP - 360

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 2

ER -

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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