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 immersed in the unit sphere with second fundamental form of constant length whose square is denoted by. Is the set of values for discrete? What is the infimum of these values of ?

The first question, i.e., whether the set of values for σ is discrete, can be reformulated as follows:
Let be a closed minimal submanifold in with the second fundamental form of constant length, denote by the set of all the possible values for the squared length of the second fundamental form of, is 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 :
Consider the set of all compact minimal hypersurfaces in 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 with constant scalar curvature. Then for each the set of all possible values for is discrete

This became known as the Chern's conjecture for minimal hypersurfaces in spheres
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 :
Let be a closed, minimally immersed hypersurface of the unit sphere with constant scalar curvature. Then is isoparametric

Here, refers to the -dimensional sphere, and n ≥ 2.
In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with taken instead of :
Let be a closed, minimally immersed submanifold in the unit sphere with constant. If, then there is a constant such that

Here, denotes an n-dimensional minimal submanifold; denotes the second largest eigenvalue of the semi-positive symmetric matrix where s are the shape operators of with respect to a given normal orthonormal frame. is rewritable as.
Another related conjecture was proposed by Robert Bryant :
A piece of a minimal hypersphere of with constant scalar curvature is isoparametric of type

Formulated alternatively:
Let be a minimal hypersurface with constant scalar curvature. Then is isoparametric

Chern's conjectures hierarchically

Put hierarchically and formulated in a single style, Chern's conjectures can look like this:
Let be a compact minimal hypersurface in the unit sphere. If has constant scalar curvature, then the possible values of the scalar curvature of form a discrete set

If has constant scalar curvature, then is isoparametric

Denote by the squared length of the second fundamental form of. Set, for. Then we have:
  • For any fixed, if, then is isoparametric, and or
  • If, then is isoparametric, and
Or alternatively:
Denote by the squared length of the second fundamental form of. Set, for. Then we have:
  • For any fixed, if, then is isoparametric, and or
  • If, then is isoparametric, and
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 be a -dimensional closed minimal hypersurface in. Does there exist a positive constant depending only on such that if, then, i.e., is one of the Clifford torus ?

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 be an -dimensional compact minimal hypersurface in. Denote by the first eigenvalue of the Laplace operator acting on functions over :
  • Is it possible to prove that if has constant scalar curvature, then ?
  • Set. Is it possible to prove that if for some, or, then ?
The second is their own generalized Chern's conjecture for hypersurfaces with constant mean curvature:
Let be a closed hypersurface with constant mean curvature in the unit sphere :
  • Assume that, where and. Is it possible to prove that or, and is an isoparametric hypersurface in ?
  • Suppose that, where. Can one show that, and is an isoparametric hypersurface in ?