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 $S^$ 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 $\sigma > \frac$?

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 $\mathbb^$ with the second fundamental form of constant length, denote by $\mathcal_n$ the set of all the possible values for the squared length of the second fundamental form of $M^n$, is $\mathcal_n$ 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 $M \subset \mathbb^$ 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 $S^$ with constant scalar curvature. Then $M$ is isoparametric

Here, $S^$ 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 $\mathbb^$ with constant $\sigma + \lambda_2$. If $\sigma + \lambda_2 > n$, then there is a constant $\epsilon(n, m) > 0$ such that$\sigma + \lambda_2 > n + \epsilon(n, m)$

Here, $M^n$ denotes an n-dimensional minimal submanifold; $\lambda_2$ denotes the second largest eigenvalue of the semi-positive symmetric matrix $S := (\left \langle A^\alpha, B^\beta \right \rangle)$ 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 $^2$.

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

A piece of a minimal hypersphere of $\mathbb^4$ with constant scalar curvature is isoparametric of type $g \le 3$

Formulated alternatively:

Let $M \subset \mathbb^4$ 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 $\mathbb^$. 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 $a_k = (k - \operatorname(5-k))n$, for $k \in \$. Then we have:
* For any fixed $k \in \$, if $a_k \le S \le a_$, then $M$ is isoparametric, and $S \equiv a_k$ or $S \equiv a_$
* 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 $a_k = (k - \operatorname(5-k))n$, for $k \in \$. Then we have:
* For any fixed $k \in \$, if $a_k \le ^2 \le a_$, then $M$ is isoparametric, and $^2 \equiv a_k$ or $^2 \equiv a_$
* If $^2 \ge a_5$, then $M$ is isoparametric, and $^2 \equiv a_5$

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 $S^, n \ge 6$. 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 $S^k\left(\sqrt\right) \times S^\left(\sqrt\right), k = 1, 2, \ldots, n-1$?

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 $\mathbb^$. 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 $a_k = (k - \operatorname(5-k))n$. Is it possible to prove that if $a_k \le S \le a_$ for some $k \in \$, 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 $\mathbb^$:

* Assume that $a \le S \le b$, where $a < b$ and \left a, b \right \cap I = \left \lbrace a, b \right \rbrace. Is it possible to prove that $S \equiv a$ or $S \equiv b$, and $M$ is an isoparametric hypersurface in $\mathbb^$?

* Suppose that $S \le c$, where $c = \sup_$. Can one show that $S \equiv c$, and $M$ is an isoparametric hypersurface in $\mathbb^$?

Sources

* 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
*
* C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
* {{cite arXiv , first1=Li , last1=Lei , first2=Hongwei , last2=Xu , first3=Zhiyuan , last3=Xu , date=2017 , title=On Chern's conjecture for minimal hypersurfaces in spheres , eprint=1712.01175 , class=math.DG

Conjectures
Unsolved problems in geometry
Differential geometry