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:

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

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):

Formulated alternatively:

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):

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 ${\displaystyle \sigma }$:

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):

Formulated alternatively:

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):

• The refined/stronger version (isoparametric hypersurfaces conjecture) of the conjecture is the same, but with the "if" part being replaced with this:

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

Or alternatively:

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

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:

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:

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

• 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].