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 {\displaystyle S^{n+1}} is n {\displaystyle n} ? If true, it will imply that the area of embedded minimal hypersurfaces in S 3 {\displaystyle 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 {\displaystyle 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 {\displaystyle M^{d}} be a closed minimal submanifold in the unit sphere S N + 1 {\displaystyle S^{N+1}} (1) with dimension d {\displaystyle d} of M d {\displaystyle M^{d}} satisfying d ≥ 2 n / 3 + 1 {\displaystyle d\geq 2n/3+1} . Is it true that the first eigenvalue of M d {\displaystyle M^{d}} is d {\displaystyle d} ?
Further reading Yau, S. T. (1982). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 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. arXiv:1201.0666. doi:10.4310/jdg/1370979337.