View source for 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 [[Biography:Shing-Tung Yau|Shing-Tung Yau]] in 1982. It asks:
<blockquote>Is it true that the first eigenvalue for the [[Laplace–Beltrami operator]] on an embedded minimal hypersurface of <math>S^{n+1}</math> is <math>n</math>?</blockquote>

If true, it will imply that the area of embedded minimal [[Hypersurface|hypersurface]]s in <math>S^3</math> will have an upper bound depending only on the [[Genus (mathematics)|genus]].

Some possible reformulations are as follows:

* <blockquote>The first eigenvalue of every closed embedded [[Minimal surface|minimal]] hypersurface <math>M^n</math> in the unit sphere <math>S^{n+1}</math>(1) is <math>n</math></blockquote>

* <blockquote>The first eigenvalue of an embedded compact minimal hypersurface <math>M^n</math> of the standard (''n''&nbsp;+&nbsp;1)-sphere with sectional curvature 1 is <math>n</math></blockquote>

* <blockquote>If <math>S^{n+1}</math> is the unit (''n''&nbsp;+&nbsp;1)-sphere with its standard round metric, then the first Laplacian eigenvalue on a closed embedded minimal hypersurface <math>{\sum}^n \subset S^{n+1}</math> is <math>n</math></blockquote>

The Yau's conjecture is verified for several special cases, but still open in general.

[[Biography:Shiing-Shen Chern|Shiing-Shen Chern]] [[Chern's conjecture for hypersurfaces in spheres|conjectured]] that a closed, minimally immersed hypersurface in <math>S^{n+1}</math>(1), whose [[Second fundamental form|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:

<blockquote>Let <math>M^d</math> be a closed minimal submanifold in the unit sphere <math>S^{N+1}</math>(1) with dimension <math>d</math> of <math>M^d</math> satisfying <math>d \ge \frac{2}{3}n + 1</math>. Is it true that the first eigenvalue of <math>M^d</math> is <math>d</math>?</blockquote>

==Further reading==
* {{cite book |first=S. T. |last=Yau |title=Seminar on Differential Geometry |series=Annals of Mathematics Studies |volume=102 |publisher=Princeton University Press |year=1982 |pages=669–706 |isbn=0-691-08268-5 }} (Problem 100)
* {{cite book |first=J. |last=Ge |first2=Z. |last2=Tang |chapter=Chern Conjecture and Isoparametric Hypersurfaces |title=Differential Geometry: Under the influence of S.S. Chern |year=2012 |publisher=Higher Education Press |location=Beijing |isbn=978-1-57146-249-7 }}
* {{cite journal |first=Z. |last=Tang |first2=W. |last2=Yan |title=Isoparametric Foliation and Yau Conjecture on the First Eigenvalue |journal=[[Journal of Differential Geometry]] |volume=94 |issue=3 |year=2013 |pages=521–540 |doi=10.4310/jdg/1370979337 |arxiv=1201.0666 }}

[[Category:Differential geometry]]
[[Category:Conjectures]]
[[Category:Unsolved problems in mathematics]]

{{Sourceattribution|Yau's conjecture on the first eigenvalue}}