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 [math]\displaystyle{ S^{n+1} }[/math] is [math]\displaystyle{ n }[/math]?

If true, it will imply that the area of embedded minimal hypersurfaces in [math]\displaystyle{ S^3 }[/math] 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 [math]\displaystyle{ M^n }[/math] in the unit sphere [math]\displaystyle{ S^{n+1} }[/math](1) is [math]\displaystyle{ n }[/math]

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

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

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

Further reading

• Yau, S. T. (1982). Seminar on Differential Geometry. Annals of Mathematics Studies. 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. doi:10.4310/jdg/1370979337.