Abstract
Suppose that M is a complete noncompact hypersurface in a sphere \(\mathbb {S}^{n+1}\) \((n\ge 3)\) with finite total curvature. We show that each p-th space of reduced \(L^2\)-cohomology on M has finite dimension, for \(0\le p\le n\). This result solves the conjecture posed in Zhu (Ann Braz Acad Sci 88:2053–2065, 2016).
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Acknowledgements
The author would like to thank professor Detang Zhou for useful discussion. The work was partially supported by NSFC Grant 11801229 and Qing Lan Project.
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Zhu, P. Hypersurfaces in Spheres with Finite Total Curvature. Results Math 74, 153 (2019). https://doi.org/10.1007/s00025-019-1082-z
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DOI: https://doi.org/10.1007/s00025-019-1082-z