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
?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 first eigenvalue of every closed embedded minimal hypersurface
The first eigenvalue of an embedded compact minimal hypersurface
of the standard (n + 1)-sphere with sectional curvature 1 is
is the unit (n + 1)-sphere with its standard round metric, then the first Laplacian eigenvalue on a closed embedded minimal hypersurface
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.
be a closed minimal submanifold in the unit sphere