Asian Journal of Mathematics

Volume 25 (2021)

Number 2

Area of minimal hypersurfaces in the unit sphere

Pages: 183 – 194

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n2.a2

Authors

Qing-Ming Cheng (Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, Fukuoka, Japan)

Guoxin Wei (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Yuting Zeng (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Abstract

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

The full text of this article is unavailable through your IP address: 44.210.77.73

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