Chern's conjecture for hypersurfaces in spheres
# 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 $\mathbb {S} ^{n+m}$ with the second fundamental form of constant length, denote by ${\mathcal {A}}_{n}$ the set of all the possible values for the squared length of the second fundamental form of $M^{n}$, is ${\mathcal {A}}_{n}$ 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 $M\subset \mathbb {S} ^{n+1}$ 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 $\mathbb {S} ^{n+m}$ with constant $\sigma +\lambda _{2}$. If $\sigma +\lambda _{2}>n$, then there is a constant $\epsilon (n,m)>0$ such that$\sigma +\lambda _{2}>n+\epsilon (n,m)$

Here, $M^{n}$ denotes an n-dimensional minimal submanifold; $\lambda _{2}$ denotes the second largest eigenvalue of the semi-positive symmetric matrix $S:=(\left\langle A^{\alpha },B^{\beta }\right\rangle )$ 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 ${\left\Vert \sigma \right\Vert }^{2}$.

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

   A piece of a minimal hypersphere of $\mathbb {S} ^{4}$ with constant scalar curvature is isoparametric of type $g\leq 3$

Formulated alternatively:

   Let $M\subset \mathbb {S} ^{4}$ 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 $\mathbb {S} ^{n+1}$. 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 $k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 5\}$. Then we have:
   * For any fixed $k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 4\}$, if $a_{k}\leq S\leq a_{k+1}$, then $M$ is isoparametric, and $S\equiv a_{k}$ or $S\equiv a_{k+1}$

   * If $S\geq a_{5}$, 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 $k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 5\}$. Then we have:
   * For any fixed $k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 4\}$, if $a_{k}\leq {\left\vert A\right\vert }^{2}\leq a_{k+1}$, 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 ${\left\vert A\right\vert }^{2}\geq a_{5}$, 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 $S^{n+1},n\geq 6$. Does there exist a positive constant $\delta (n)$ depending only on $n$ such that if $n\leq n+\delta (n)$, then $S\equiv n$, i.e., $M$ is one of the Clifford torus $S^{k}\left({\sqrt {\frac {k}{n}}}\right)\times S^{n-k}\left({\sqrt {\frac {n-k}{n}}}\right),k=1,2,\ldots ,n-1$?

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 $\mathbb {S} ^{n+1}$. 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 $a_{k}=(k-\operatorname {sgn} (5-k))n$. Is it possible to prove that if $a_{k}\leq S\leq a_{k+1}$ for some $k\in \{m\in \mathbb {Z} ^{+};2\leq m\leq 4\}$, or $S\geq a_{5}$, 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 $\mathbb {S} ^{n+1}$:
   * Assume that $a\leq S\leq b$, 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 $\mathbb {S} ^{n+1}$?

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

## 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. arXiv:0803.0502v3. doi:10.1016/j.jfa.2011.05.002. S2CID 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
* Lei, Li; Xu, Hongwei; Xu, Zhiyuan (2017). "On Chern's conjecture for minimal hypersurfaces in spheres". arXiv:1712.01175 [math.DG].
