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 $M^n$ 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 $\sigma >\frac{n}{2-\frac{1}{m}}$?

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

Let $M^n$ be a closed minimal submanifold in ${𝕊}^{n+m}$ with the second fundamental form of constant length, denote by ${𝒜}_n$ the set of all the possible values for the squared length of the second fundamental form of $M^n$, is ${𝒜}_n$ 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 $\sigma +{\lambda }_2$ taken instead of $\sigma$:

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

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

Another related conjecture was proposed by Robert Bryant:

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 $a_k=(k-\mathrm{sgn}(5-k))n$, for $k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}$. Then we have:

• For any fixed $k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 4\}$, 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 $a_k=(k-\mathrm{sgn}(5-k))n$, for $k\in \{m\in {\mathbb{Z}}^{+};1\le m\le 5\}$. Then we have:

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

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

Other related and still open problems

Besides the conjectures of Lu and Bryant, there are 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 $S^k\left(\sqrt{\frac{k}{n}}\right)\times S^{n-k}\left(\sqrt{\frac{n-k}{n}}\right),k=1,2,\dots ,n-1$?

In 2017, Li Lei, Hongwei Xu and Zhiyuan Xu proposed two Chern-related problems.

The first 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 ${\lambda }_1(M)$ 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 ${\lambda }_1(M)=n$?

• Set $a_k=(k-\mathrm{sgn}(5-k))n$. Is it possible to prove that if $a_k\le S\le a_{k+1}$ for some $k\in \{m\in {\mathbb{Z}}^{+};2\le m\le 4\}$, or $S\ge a_5$, then ${\lambda }_1(M)=n$?

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 $[a,b]\cap I=\{a,b\}$. 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 $c=\underset{t\in I}{sup}t$. 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
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• C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
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