L2-Harmonic forms on incomplete riemannian manifolds with positive Ricci curvature

Junya Takahashi

doi:10.3390/math6050075

L²-Harmonic forms on incomplete riemannian manifolds with positive Ricci curvature

Junya Takahashi

INF - System Information Sciences

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

Abstract

We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L²-harmonic forms and on which the L²-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.

Original language	English
Article number	75
Journal	Mathematics
Volume	6
Issue number	5
DOIs	https://doi.org/10.3390/math6050075
Publication status	Published - 2018 May 9

Keywords

Capacity
Hodge-Laplacian
L-Stokes theorem
L-harmonic forms
Manifold with singularity

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.3390/math6050075

Cite this

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title = "L2-Harmonic forms on incomplete riemannian manifolds with positive Ricci curvature",

abstract = "We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L2-harmonic forms and on which the L2-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.",

keywords = "Capacity, Hodge-Laplacian, L-Stokes theorem, L-harmonic forms, Manifold with singularity",

author = "Junya Takahashi",

note = "Funding Information: The author is grateful to Jun Masamune for valuable discussion. The author is also grateful to the referees for helpful comments. The author is supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 16K05117. Publisher Copyright: {\textcopyright} 2018 by the author.",

year = "2018",

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AU - Takahashi, Junya

N1 - Funding Information: The author is grateful to Jun Masamune for valuable discussion. The author is also grateful to the referees for helpful comments. The author is supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 16K05117. Publisher Copyright: © 2018 by the author.

PY - 2018/5/9

Y1 - 2018/5/9

N2 - We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L2-harmonic forms and on which the L2-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.

AB - We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L2-harmonic forms and on which the L2-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.

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KW - Hodge-Laplacian

KW - L-Stokes theorem

KW - L-harmonic forms

KW - Manifold with singularity

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