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 [math]\displaystyle{ M^n }[/math] immersed in the unit sphere [math]\displaystyle{ S^{n+m} }[/math] with second fundamental form of constant length whose square is denoted by [math]\displaystyle{ \sigma }[/math]. Is the set of values for [math]\displaystyle{ \sigma }[/math] discrete? What is the infimum of these values of [math]\displaystyle{ \sigma \gt \frac{n}{2-\frac{1}{m}} }[/math]?

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

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

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

In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with [math]\displaystyle{ \sigma + \lambda_2 }[/math] taken instead of [math]\displaystyle{ \sigma }[/math]:

Let [math]\displaystyle{ M^n }[/math] be a closed, minimally immersed submanifold in the unit sphere [math]\displaystyle{ \mathbb{S}^{n+m} }[/math] with constant [math]\displaystyle{ \sigma + \lambda_2 }[/math]. If [math]\displaystyle{ \sigma + \lambda_2 \gt n }[/math], then there is a constant [math]\displaystyle{ \epsilon(n, m) \gt 0 }[/math] such that[math]\displaystyle{ \sigma + \lambda_2 \gt n + \epsilon(n, m) }[/math]

Here, [math]\displaystyle{ M^n }[/math] denotes an n-dimensional minimal submanifold; [math]\displaystyle{ \lambda_2 }[/math] denotes the second largest eigenvalue of the semi-positive symmetric matrix [math]\displaystyle{ S := (\left \langle A^\alpha, B^\beta \right \rangle) }[/math] where [math]\displaystyle{ A^\alpha }[/math]s ([math]\displaystyle{ \alpha = 1, \cdots, m }[/math]) are the shape operators of [math]\displaystyle{ M }[/math] with respect to a given (local) normal orthonormal frame. [math]\displaystyle{ \sigma }[/math] is rewritable as [math]\displaystyle{ {\left \Vert \sigma \right \Vert}^2 }[/math].

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

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

Formulated alternatively:

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

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

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

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

Or alternatively:

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

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

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 [math]\displaystyle{ M }[/math] be a [math]\displaystyle{ n }[/math]-dimensional closed minimal hypersurface in [math]\displaystyle{ S^{n+1}, n \ge 6 }[/math]. Does there exist a positive constant [math]\displaystyle{ \delta(n) }[/math] depending only on [math]\displaystyle{ n }[/math] such that if [math]\displaystyle{ n \le n + \delta(n) }[/math], then [math]\displaystyle{ S \equiv n }[/math], i.e., [math]\displaystyle{ M }[/math] is one of the Clifford torus [math]\displaystyle{ 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 }[/math]?

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 [math]\displaystyle{ M }[/math] be an [math]\displaystyle{ n }[/math]-dimensional compact minimal hypersurface in [math]\displaystyle{ \mathbb{S}^{n+1} }[/math]. Denote by [math]\displaystyle{ \lambda_1(M) }[/math] the first eigenvalue of the Laplace operator acting on functions over [math]\displaystyle{ M }[/math]:

• Is it possible to prove that if [math]\displaystyle{ M }[/math] has constant scalar curvature, then [math]\displaystyle{ \lambda_1(M) = n }[/math]?

• Set [math]\displaystyle{ a_k = (k - \operatorname{sgn}(5-k))n }[/math]. Is it possible to prove that if [math]\displaystyle{ a_k \le S \le a_{k+1} }[/math] for some [math]\displaystyle{ k \in \{ m \in \mathbb{Z}^+ ; 2 \le m \le 4 \} }[/math], or [math]\displaystyle{ S \ge a_5 }[/math], then [math]\displaystyle{ \lambda_1(M) = n }[/math]?

The second is their own generalized Chern's conjecture for hypersurfaces with constant mean curvature:

Let [math]\displaystyle{ M }[/math] be a closed hypersurface with constant mean curvature [math]\displaystyle{ H }[/math] in the unit sphere [math]\displaystyle{ \mathbb{S}^{n+1} }[/math]:

• Assume that [math]\displaystyle{ a \le S \le b }[/math], where [math]\displaystyle{ a \lt b }[/math] and [math]\displaystyle{ \left [ a, b \right ] \cap I = \left \lbrace a, b \right \rbrace }[/math]. Is it possible to prove that [math]\displaystyle{ S \equiv a }[/math] or [math]\displaystyle{ S \equiv b }[/math], and [math]\displaystyle{ M }[/math] is an isoparametric hypersurface in [math]\displaystyle{ \mathbb{S}^{n+1} }[/math]?

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

Sources

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