$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature
Year: 2021
Author: Jiuru Zhou
Journal of Mathematical Study, Vol. 54 (2021), Iss. 4 : pp. 396–406
Abstract
In general, the space of $L^p$ harmonic forms $\mathcal{H}^k(L^p(M))$ and reduced $L^p$ cohomology $H^k(L^p(M))$ might be not isomorphic on a complete Riemannian manifold $M$, except for $p=2$. Nevertheless, one can consider whether $\mathrm{dim}\mathcal{H}^k(L^p(M))<+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))<+\infty$. In order to study such kind of problems, this paper obtains that dimension of space of $L^p$ harmonic forms on a hypersurface in unit sphere with finite total curvature is finite, which is also a generalization of the previous work by Zhu. The next step will be the investigation of dimension of the reduced $L^p$ cohomology on such hypersurfaces.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v54n4.21.05
Journal of Mathematical Study, Vol. 54 (2021), Iss. 4 : pp. 396–406
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: $L^p$ harmonic $k$-form hypersurface in sphere total curvature.