Chern's conjecture for hypersurfaces in spheres

{{Short description|Ugandan Social Media influencer / blogger born 1995 in mbarara town}}{{inline|date=March 2019}}

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 $g \le 3$

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 $k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 5 \}$ . 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 $a_k = (k - \operatorname{sgn}(5-k))n$ , 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 $a_k \le {\left \vert A \right \vert}^2 \le a_{k+1}$ , 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.

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
{{cite journal |first=Zhiqin |last=Lu |date=2011 |title=Normal Scalar Curvature Conjecture and its applications |journal=Journal of Functional Analysis |volume=261 |issue=5 |pages=1284–1308 |doi=10.1016/j.jfa.2011.05.002 |arxiv=0803.0502v3 |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
{{cite arXiv |first1=Li |last1=Lei |first2=Hongwei |last2=Xu |first3=Zhiyuan |last3=Xu |date=2017 |title=On Chern's conjecture for minimal hypersurfaces in spheres |eprint=1712.01175 |class=math.DG}}

Category:Conjectures

Category:Unsolved problems in geometry

Category:Differential geometry

Chern's conjecture for hypersurfaces in spheres

Chern's conjectures hierarchically

Other related and still open problems

References