Chern's conjecture for hypersurfaces in spheres

Consider closed minimal submanifolds ${\displaystyle M^n}$ immersed in the unit sphere ${\displaystyle S^{n+m}}$ with second fundamental form of constant length whose square is denoted by ${\displaystyle \sigma }$. Is the set of values for ${\displaystyle \sigma }$ discrete? What is the infimum of these values of ${\displaystyle \sigma >\frac{n}{2-\frac{1}{m}}}$?

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

Let ${\displaystyle M^n}$ be a closed minimal submanifold in ${\displaystyle {\mathbb{S}}^{n+m}}$ with the second fundamental form of constant length, denote by ${\displaystyle {\mathcal{A}}_n}$ the set of all the possible values for the squared length of the second fundamental form of ${\displaystyle M^n}$, is ${\displaystyle {\mathcal{A}}_n}$ 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 ${\displaystyle S^N}$ 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 ${\displaystyle M\subset {\mathbb{S}}^{n+1}}$ with constant scalar curvature ${\displaystyle k}$. Then for each ${\displaystyle n}$ the set of all possible values for ${\displaystyle k}$ (or equivalently ${\displaystyle S}$) 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 ${\displaystyle M^n}$ be a closed, minimally immersed hypersurface of the unit sphere ${\displaystyle S^{n+1}}$ with constant scalar curvature. Then ${\displaystyle M}$ is isoparametric

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

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

Let ${\displaystyle M^n}$ be a closed, minimally immersed submanifold in the unit sphere ${\displaystyle {\mathbb{S}}^{n+m}}$ with constant ${\displaystyle \sigma +{\lambda }_2}$. If ${\displaystyle \sigma +{\lambda }_2>n}$, then there is a constant ${\displaystyle ϵ(n,m)>0}$ such that${\displaystyle \sigma +{\lambda }_2>n+ϵ(n,m)}$

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

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

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

Formulated alternatively:

Let ${\displaystyle M\subset {\mathbb{S}}^4}$ be a minimal hypersurface with constant scalar curvature. Then ${\displaystyle M}$ 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 ${\displaystyle M}$ be a compact minimal hypersurface in the unit sphere ${\displaystyle {\mathbb{S}}^{n+1}}$. If ${\displaystyle M}$ has constant scalar curvature, then the possible values of the scalar curvature of ${\displaystyle M}$ 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 ${\displaystyle M}$ has constant scalar curvature, then ${\displaystyle M}$ is isoparametric

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

Denote by ${\displaystyle S}$ the squared length of the second fundamental form of ${\displaystyle M}$. Set ${\displaystyle a_k=(k-\mathrm{sgn}(5-k))n}$, for ${\displaystyle k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}}$. Then we have:

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

Or alternatively:

Denote by ${\displaystyle A}$ the squared length of the second fundamental form of ${\displaystyle M}$. Set ${\displaystyle a_k=(k-\mathrm{sgn}(5-k))n}$, for ${\displaystyle k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}}$. Then we have:

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

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