Yau's conjecture on the first eigenvalue

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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 is ?

If true, it will imply that the area of embedded minimal hypersurfaces in will have an upper bound depending only on the genus.

Some possible reformulations are as follows:

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 (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 be a closed minimal submanifold in the unit sphere (1) with dimension of satisfying . Is it true that the first eigenvalue of is ?

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