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 $S^{n+1}$ 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.

Some possible reformulations are as follows:


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


 * "The first eigenvalue of an embedded compact minimal hypersurface $M^n$ of the standard (n + 1)-sphere with sectional curvature 1 is $n$"


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

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 $$S^{n+1}$$(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 $d \ge \frac{2}{3}n + 1$. Is it true that the first eigenvalue of $M^d$ is $d$?"