Yau's conjecture on the first eigenvalue

In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks:

Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of ${\displaystyle S^{n+1}}$ is ${\displaystyle n}$?

If true, it will imply that the area of embedded minimal hypersurfaces in ${\displaystyle S^{3}}$ will have an upper bound depending only on the genus.

Some possible reformulations are as follows:

• The first eigenvalue of every closed embedded minimal hypersurface ${\displaystyle M^{n}}$ in the unit sphere ${\displaystyle S^{n+1}}$(1) is ${\displaystyle n}$

• The first eigenvalue of an embedded compact minimal hypersurface ${\displaystyle M^{n}}$ of the standard (n + 1)-sphere with sectional curvature 1 is ${\displaystyle n}$

• If ${\displaystyle S^{n+1}}$ is the unit (n + 1)-sphere with its standard round metric, then the first Laplacian eigenvalue on a closed embedded minimal hypersurface ${\displaystyle {\sum }^{n}\subset S^{n+1}}$ is ${\displaystyle n}$

The Yau's conjecture is verified for several special cases, but still open in general.

Shiing-Shen Chern conjectured that a closed, minimally immersed hypersurface in ${\displaystyle S^{n+1}}$(1), whose second fundamental form has constant length, is isoparametric. If true, it would have established the Yau's conjecture for the minimal hypersurface whose second fundamental form has constant length.

A possible generalization of the Yau's conjecture:

Let ${\displaystyle M^{d}}$ be a closed minimal submanifold in the unit sphere ${\displaystyle S^{N+1}}$(1) with dimension ${\displaystyle d}$ of ${\displaystyle M^{d}}$ satisfying ${\displaystyle d\geq {\frac {2}{3}}n+1}$. Is it true that the first eigenvalue of ${\displaystyle M^{d}}$ is ${\displaystyle d}$?

• Yau, S. T. (1982). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton University Press. pp. 669–706. ISBN 0-691-08268-5. (Problem 100)
• Ge, J.; Tang, Z. (2012). "Chern Conjecture and Isoparametric Hypersurfaces". Differential Geometry: Under the influence of S.S. Chern. Beijing: Higher Education Press. ISBN 978-1-57146-249-7.
• Tang, Z.; Yan, W. (2013). "Isoparametric Foliation and Yau Conjecture on the First Eigenvalue". Journal of Differential Geometry. 94 (3): 521–540. arXiv:1201.0666. doi:10.4310/jdg/1370979337.