Area of minimal hypersurfaces in the unit sphere

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k (\sqrt{\frac{k}{n}}) \times S^{n-k} (\sqrt{\frac{n-k}{n}})$ gives the lowest value of area among all non-totally geodesic compact minimal hypersurfaces in the unit sphere $S^{n+1} (1)$. The present paper shows that Yau conjecture is true for minimal rotational hypersurfaces, more precisely, the area $\lvert M^n \rvert$ of compact minimal rotational hypersurface $M^n$ is either equal to $\lvert S^n (1) \rvert$, or equal to $\lvert S^1 (\sqrt{\frac{1}{n}}) \times S^{n-1} (\sqrt{\frac{n-1}{n}}) \rvert$ or greater than $2 (1-\frac{1}{\pi}) \lvert S^1 \sqrt{\frac{1}{n}} \times S^n - 1(\sqrt{\frac{n-1}{n}}) \rvert$ As the application, the entropies of some special self-shrinkers are estimated.

Keywords

minimal hypersurfaces, Yau conjecture, area, self-shrinkers, entropy

2010 Mathematics Subject Classification

53A10, 53C42

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The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (B): No. 16H03937.

The second author was partly supported by NSFC Grant No. 11771154, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2018), Guangdong Natural Science Foundation Grant No. 2019A1515011451.

Received 6 July 2017

Accepted 18 June 2020

Published 15 October 2021