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 $\sigma > \frac{n}{2-\frac{1}{m}}$?|undefined"

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 $\mathbb{S}^{n+m}$ with the second fundamental form of constant length, denote by $\mathcal{A}_n$ the set of all the possible values for the squared length of the second fundamental form of $M^n$, is $\mathcal{A}_n$ 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 $M \subset \mathbb{S}^{n+1}$ 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 $\mathbb{S}^{n+m} $ 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 $${\left \Vert \sigma \right \Vert}^2$$.

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

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

Formulated alternatively:

"Let $M \subset \mathbb{S}^4$ 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 $\mathbb{S}^{n+1}$. 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 $$a_k = (k - \operatorname{sgn}(5-k))n$$, 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 $$a_k = (k - \operatorname{sgn}(5-k))n$$, 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 $$a_k \le {\left \vert A \right \vert}^2 \le a_{k+1}$$, then $$M$$ is isoparametric, and $${\left \vert A \right \vert}^2 \equiv a_k$$ or $${\left \vert A \right \vert}^2 \equiv a_{k+1}$$
 * If $${\left \vert A \right \vert}^2 \ge a_5$$, then $$M$$ is isoparametric, and $${\left \vert A \right \vert}^2 \equiv a_5$$

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 $S^k\left(\sqrt{\frac{k}{n}}\right) \times S^{n-k}\left(\sqrt{\frac{n-k}{n}}\right), k = 1, 2, \ldots, n-1$?|undefined"

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 $$\mathbb{S}^{n+1}$$. 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 $$a_k = (k - \operatorname{sgn}(5-k))n$$. 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 $$\mathbb{S}^{n+1}$$:


 * 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 $$\mathbb{S}^{n+1}$$?


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