Chern's conjecture for hypersurfaces in spheres

Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question:

Consider closed minimal submanifolds

$M^n$

immersed in the unit sphere

$S^{n+m}$

with second fundamental form of constant length whose square is denoted by

$\sigma$

. Is the set of values for

$\sigma$

discrete? What is the infimum of these values of

$\sigma >\frac{n}{2-\frac{1}{m}}$

?

The first question, i.e., whether the set of values for σ is discrete, can be reformulated as follows:

Let

$M^n$

be a closed minimal submanifold in

${𝕊}^{n+m}$

with the second fundamental form of constant length, denote by

${𝒜}_n$

the set of all the possible values for the squared length of the second fundamental form of

$M^n$

, is

${𝒜}_n$

a discrete?

Its affirmative hand, more general than the Chern's conjecture for hypersurfaces, sometimes also referred to as the Chern's conjecture and is still, as of 2018, unanswered even with M as a hypersurface (Chern proposed this special case to the Shing-Tung Yau's open problems' list in differential geometry in 1982):

Consider the set of all compact minimal hypersurfaces in

$S^N$

with constant scalar curvature. Think of the scalar curvature as a function on this set. Is the image of this function a discrete set of positive numbers?

Formulated alternatively:

Consider closed minimal hypersurfaces

$M\subset {𝕊}^{n+1}$

with constant scalar curvature

$k$

. Then for each

$n$

the set of all possible values for

$k$

(or equivalently

$S$

) is discrete

This became known as the Chern's conjecture for minimal hypersurfaces in spheres (or Chern's conjecture for minimal hypersurfaces in a sphere)

This hypersurface case was later, thanks to progress in isoparametric hypersurfaces' studies, given a new formulation, now known as Chern's conjecture for isoparametric hypersurfaces in spheres (or Chern's conjecture for isoparametric hypersurfaces in a sphere):

Let

$M^n$

be a closed, minimally immersed hypersurface of the unit sphere

$S^{n+1}$

with constant scalar curvature. Then

$M$

is isoparametric

Here, $S^{n+1}$ refers to the (n+1)-dimensional sphere, and n ≥ 2.

In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with $\sigma +{\lambda }_2$ taken instead of $\sigma$:

Let

$M^n$

be a closed, minimally immersed submanifold in the unit sphere

${𝕊}^{n+m}$

with constant

$\sigma +{\lambda }_2$

. If

$\sigma +{\lambda }_2>n$

, then there is a constant

$ϵ(n,m)>0$

such that

$\sigma +{\lambda }_2>n+ϵ(n,m)$

Here, $M^n$ denotes an n-dimensional minimal submanifold; ${\lambda }_2$ denotes the second largest eigenvalue of the semi-positive symmetric matrix $S:=(⟨A^{\alpha },B^{\beta }⟩)$ where $A^{\alpha }$s ($\alpha =1,\cdots ,m$) are the shape operators of $M$ with respect to a given (local) normal orthonormal frame. $\sigma$ is rewritable as ${\Vert \sigma \Vert }^2$.

Another related conjecture was proposed by Robert Bryant (mathematician):

A piece of a minimal hypersphere of

${𝕊}^4$

with constant scalar curvature is isoparametric of type

$g\le 3$

Formulated alternatively:

Let

$M\subset {𝕊}^4$

be a minimal hypersurface with constant scalar curvature. Then

$M$

is isoparametric

Chern's conjectures hierarchically

Put hierarchically and formulated in a single style, Chern's conjectures (without conjectures of Lu and Bryant) can look like this:

• The first version (minimal hypersurfaces conjecture):

Let

$M$

be a compact minimal hypersurface in the unit sphere

${𝕊}^{n+1}$

. If

$M$

has constant scalar curvature, then the possible values of the scalar curvature of

$M$

form a discrete set

• The refined/stronger version (isoparametric hypersurfaces conjecture) of the conjecture is the same, but with the "if" part being replaced with this:

If

$M$

has constant scalar curvature, then

$M$

is isoparametric

• The strongest version replaces the "if" part with:

Denote by

$S$

the squared length of the second fundamental form of

$M$

. Set

$a_k=(k-\mathrm{sgn}(5-k))n$

, for

$k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}$

. Then we have:

• For any fixed $k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 4\}$, if $a_k\le S\le a_{k+1}$, then $M$ is isoparametric, and $S\equiv a_k$ or $S\equiv a_{k+1}$
• If $S\ge a_5$, then $M$ is isoparametric, and $S\equiv a_5$

Or alternatively:

Denote by

$A$

the squared length of the second fundamental form of

$M$

. Set

$a_k=(k-\mathrm{sgn}(5-k))n$

, for

$k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}$

. Then we have:

• For any fixed $k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 4\}$, if $a_k\le {\left|A\right|}^2\le a_{k+1}$, then $M$ is isoparametric, and ${\left|A\right|}^2\equiv a_k$ or ${\left|A\right|}^2\equiv a_{k+1}$
• If ${\left|A\right|}^2\ge a_5$, then $M$ is isoparametric, and ${\left|A\right|}^2\equiv a_5$

One should pay attention to the so-called first and second pinching problems as special parts for Chern.

Other related and still open problems

Besides the conjectures of Lu and Bryant, there're also others:

In 1983, Chia-Kuei Peng and Chuu-Lian Terng proposed the problem related to Chern:

Let

$M$

be a

$n$

-dimensional closed minimal hypersurface in

$S^{n+1},n\ge 6$

. Does there exist a positive constant

$\delta (n)$

depending only on

$n$

such that if

$n\le n+\delta (n)$

, then

$S\equiv n$

, i.e.,

$M$

is one of the Clifford torus

$S^k\left(\sqrt{\frac{k}{n}}\right)\times S^{n-k}\left(\sqrt{\frac{n-k}{n}}\right),k=1,2,\dots ,n-1$

?

In 2017, Li Lei, Hongwei Xu and Zhiyuan Xu proposed 2 Chern-related problems.

The 1st one was inspired by Yau's conjecture on the first eigenvalue:

Let

$M$

be an

$n$

-dimensional compact minimal hypersurface in

${𝕊}^{n+1}$

. Denote by

${\lambda }_1(M)$

the first eigenvalue of the Laplace operator acting on functions over

$M$

:

• Is it possible to prove that if $M$ has constant scalar curvature, then ${\lambda }_1(M)=n$?

• Set $a_k=(k-\mathrm{sgn}(5-k))n$. Is it possible to prove that if $a_k\le S\le a_{k+1}$ for some $k\in \{m\in {\mathbb{Z}}^{+};2\le m\le 4\}$, or $S\ge a_5$, then ${\lambda }_1(M)=n$?

The second is their own generalized Chern's conjecture for hypersurfaces with constant mean curvature:

Let

$M$

be a closed hypersurface with constant mean curvature

$H$

in the unit sphere

${𝕊}^{n+1}$

:

• Assume that $a\le S\le b$, where $a<b$ and $[a,b]\cap I=\{a,b\}$. Is it possible to prove that $S\equiv a$ or $S\equiv b$, and $M$ is an isoparametric hypersurface in ${𝕊}^{n+1}$?

• Suppose that $S\le c$, where $c=\underset{t\in I}{\sup }t$. Can one show that $S\equiv c$, and $M$ is an isoparametric hypersurface in ${𝕊}^{n+1}$?

References

• S.S. Chern, Minimal Submanifolds in a Riemannian Manifold, (mimeographed in 1968), Department of Mathematics Technical Report 19 (New Series), University of Kansas, 1968
• S.S. Chern, Brief survey of minimal submanifolds, Differentialgeometrie im Großen, volume 4 (1971), Mathematisches Forschungsinstitut Oberwolfach, pp. 43–60
• S.S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields: Proceedings of a Conference in honor of Professor Marshall Stone, held at the University of Chicago, May 1968 (1970), Springer-Verlag, pp. 59-75
• S.T. Yau, Seminar on Differential Geometry (Annals of Mathematics Studies, Volume 102), Princeton University Press (1982), pp. 669–706, problem 105
• L. Verstraelen, Sectional curvature of minimal submanifolds, Proceedings of the Workshop on Differential Geometry (1986), University of Southampton, pp. 48–62
• M. Scherfner and S. Weiß, Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres, Süddeutsches Kolloquium über Differentialgeometrie, volume 33 (2008), Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, pp. 1–13
• Z. Lu, Normal scalar curvature conjecture and its applications, Journal of Functional Analysis, volume 261 (2011), pp. 1284–1308
• Template:Cite journal
• C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
• Template:Cite arXiv