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L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature

Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tôhoku University, 6-3-09, Aoba, Sendai 980-8579, Japan
Dedicated to the Memory of Professor Ahmad El Soufi.
Mathematics 2018, 6(5), 75; https://doi.org/10.3390/math6050075
Received: 28 February 2018 / Revised: 24 April 2018 / Accepted: 29 April 2018 / Published: 9 May 2018
(This article belongs to the Special Issue Differential Geometry)
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Abstract

We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L 2 -harmonic forms and on which the L 2 -Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds. View Full-Text
Keywords: L2-harmonic forms; Hodge–Laplacian; manifold with singularity; L2-Stokes theorem; capacity L2-harmonic forms; Hodge–Laplacian; manifold with singularity; L2-Stokes theorem; capacity
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Takahashi, J. L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature. Mathematics 2018, 6, 75.

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