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 closed minimal hypersurface of $S^{n+1}$ is $n$?

If true, it will imply that the area of embedded minimal hypersurfaces in $S^3$ will have an upper bound depending only on the genus. 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 $S^{n+1}$(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 $M^d$ be a closed minimal submanifold in the unit sphere $S^{N+1}$(1) with dimension $d$ of $M^d$ satisfying $d\ge 2n/3+1$. Is it true that the first eigenvalue of $M^d$ is $d$?

Further reading

• Template:Cite book (Problem 100)
• Template:Cite book
• Template:Cite journal