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
with second fundamental form of constant length whose square is denoted by
?The first question, i.e., whether the set of values for σ is discrete, can be reformulated as follows: Let
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
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
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
) is discreteThis 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
be a closed, minimally immersed hypersurface of the unit sphere
In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with
be a closed, minimally immersed submanifold in the unit sphere
denotes an n-dimensional minimal submanifold;
denotes the second largest eigenvalue of the semi-positive symmetric matrix
with respect to a given (local) normal orthonormal frame.
Another related conjecture was proposed by Robert Bryant (mathematician): A piece of a minimal hypersphere of
with constant scalar curvature is isoparametric of type
be a minimal hypersurface with constant scalar curvature.
is isoparametricPut hierarchically and formulated in a single style, Chern's conjectures (without conjectures of Lu and Bryant) can look like this: Let
be a compact minimal hypersurface in the unit sphere
the squared length of the second fundamental form of
the squared length of the second fundamental form of
Then we have: One should pay attention to the so-called first and second pinching problems as special parts for Chern.
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
?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
the first eigenvalue of the Laplace operator acting on functions over
: The second is their own generalized Chern's conjecture for hypersurfaces with constant mean curvature: Let
be a closed hypersurface with constant mean curvature