Chern's conjecture for hypersurfaces in spheres explained

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 > (n)/(2- (1)/(m)) ?

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 S ^{n+m} with the second fundamental form of constant length, denote by l{A} _{n} the set of all the possible values for the squared length of the second fundamental form of M ^{n} , is l{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 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 + λ _{2} taken instead of \sigma :

Let

M ^{n} be a closed, minimally immersed submanifold in the unit sphere S ^{n+m} with constant \sigma + λ _{2} . If \sigma + λ _{2} > n , then there is a constant \epsilon(n, m) > 0 such that\sigma + λ _{2} > n + \epsilon(n, m)

Here,

M ^{n} denotes an n-dimensional minimal submanifold; λ _{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 , … , 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

S ^{4} with constant scalar curvature is isoparametric of type g \le 3

Formulated alternatively:

Let

M \subset 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 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 Z ^{+} ; 1 \le m \le 5 \} . Then we have:

• For any fixed

k \in \{ m \in 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 Z ^{+} ; 1 \le m \le 5 \} . Then we have:

• For any fixed

k \in \{ m \in 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{ (k)/(n)} }\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 S ^{n+1} . Denote by λ _{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 λ _{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 Z ^{+} ; 2 \le m \le 4 \} , or S \ge a _{5} , then λ _{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 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 S ^{n+1} ?

• Suppose that

S \le c , where c = \sup _{t} {t} . Can one show that S \equiv c , and M is an isoparametric hypersurface in S ^{n+1} ?

References

• 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
• Zhiqin . Lu . 2011 . Normal Scalar Curvature Conjecture and its applications . Journal of Functional Analysis . 261 . 5 . 1284–1308 . 10.1016/j.jfa.2011.05.002 . 0803.0502v3 . 17541544 .
• C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
• Li . Lei . Hongwei . Xu . Zhiyuan . Xu . 2017 . On Chern's conjecture for minimal hypersurfaces in spheres . 1712.01175 . math.DG.