TY - JOUR
T1 - $L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature
AU - Zhou , Jiuru
JO - Journal of Mathematical Study
VL - 4
SP - 396
EP - 406
PY - 2021
DA - 2021/06
SN - 54
DO - http://doi.org/10.4208/jms.v54n4.21.05
UR - https://global-sci.org/intro/article_detail/jms/19292.html
KW - $L^p$ harmonic $k$-form, hypersurface in sphere, total curvature.
AB - <p style="text-align: justify;">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 $\mathrm{dim}\mathcal{H}^k(L^p(M))&lt;+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))&lt;+\infty$. 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.</p>