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

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

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

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

In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with ${\displaystyle \sigma +\lambda _{2}}$ taken instead of ${\displaystyle \sigma }$:

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

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

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

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

Formulated alternatively:

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

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

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

• For any fixed ${\displaystyle k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 4\}}$, if ${\displaystyle a_{k}\leq S\leq a_{k+1}}$, then ${\displaystyle M}$ is isoparametric, and ${\displaystyle S\equiv a_{k}}$ or ${\displaystyle S\equiv a_{k+1}}$
• If ${\displaystyle S\geq a_{5}}$, then ${\displaystyle M}$ is isoparametric, and ${\displaystyle S\equiv a_{5}}$

Or alternatively:

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

• For any fixed ${\displaystyle k\in \{m\in \mathbb {Z} ^{+};1\leq m\leq 4\}}$, if ${\displaystyle a_{k}\leq {\left\vert A\right\vert }^{2}\leq a_{k+1}}$, then ${\displaystyle M}$ is isoparametric, and ${\displaystyle {\left\vert A\right\vert }^{2}\equiv a_{k}}$ or ${\displaystyle {\left\vert A\right\vert }^{2}\equiv a_{k+1}}$
• If ${\displaystyle {\left\vert A\right\vert }^{2}\geq a_{5}}$, then ${\displaystyle M}$ is isoparametric, and ${\displaystyle {\left\vert A\right\vert }^{2}\equiv a_{5}}$

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 ${\displaystyle M}$ be a ${\displaystyle n}$-dimensional closed minimal hypersurface in ${\displaystyle S^{n+1},n\geq 6}$. Does there exist a positive constant ${\displaystyle \delta (n)}$ depending only on ${\displaystyle n}$ such that if ${\displaystyle n\leq n+\delta (n)}$, then ${\displaystyle S\equiv n}$, i.e., ${\displaystyle M}$ is one of the Clifford torus ${\displaystyle S^{k}\left({\sqrt {\frac {k}{n}}}\right)\times S^{n-k}\left({\sqrt {\frac {n-k}{n}}}\right),k=1,2,\ldots ,n-1}$?

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

• Is it possible to prove that if ${\displaystyle M}$ has constant scalar curvature, then ${\displaystyle \lambda _{1}(M)=n}$?

• Set ${\displaystyle a_{k}=(k-\operatorname {sgn} (5-k))n}$. Is it possible to prove that if ${\displaystyle a_{k}\leq S\leq a_{k+1}}$ for some ${\displaystyle k\in \{m\in \mathbb {Z} ^{+};2\leq m\leq 4\}}$, or ${\displaystyle S\geq a_{5}}$, then ${\displaystyle \lambda _{1}(M)=n}$?

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

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

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

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

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
• 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].