Isoparametric Hypersurfaces in Sn+1: The Chern Conjecture

The ChernConjectureBasicsTheConjectureResultsGeneralizationsIsoparametric Hypersurfaces in S n+1 :The Chern ConjectureSummaryOutlookMike ScherfnerInstitute of Mathematics, TU Berlin

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlook1 Basics2 The Conjecture3 Results4 Generalizations5 Summary6 Outlook

ContentThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookWe present a short history of the Chern conjecture forisoparametric hypersurfaces in spheres and generalizations.Main results will be presented and we summarize the progressfor this topic.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlook

BasicsThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookWe are concerned with hypersurfaces, i. e. n-dimensionalsubmanifolds in (n + 1)-dimensional Riemannian manifolds.The curvature of a hypersurface M in the ambient manifold isdescribed by the second fundamental form h (or the associatedshape operator A). The eigenvalues of h are the principalcurvature functions λ i , i = 1...n.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookA hypersurface in a space of constant curvature c is calledisoparametric, if all the λ i are constant functions.In R n+1 all the isoparametric hypersurfaces are hyperspheres,hyperplanes or (generalized) cylinders S k × R n−k . In the sphereS n+1 we could expect “much more” of such hypersurfaces.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookA hypersurface in a space of constant curvature c is calledisoparametric, if all the λ i are constant functions.In R n+1 all the isoparametric hypersurfaces are hyperspheres,hyperplanes or (generalized) cylinders S k × R n−k . In the sphereS n+1 we could expect “much more” of such hypersurfaces.

Important QuantitiesThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe scalar curvature κ comes directly form the curvaturetensor, describing the intrinsic curvature of M and we haveκ = c +1n(n − 1)∑λ i λ j .i≠jThe mean curvature H of M is given byH = 1 ∑λ i ,niand M is minimally immersed iff H = 0.

Important QuantitiesThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe scalar curvature κ comes directly form the curvaturetensor, describing the intrinsic curvature of M and we haveκ = c +1n(n − 1)∑λ i λ j .i≠jThe mean curvature H of M is given byH = 1 ∑λ i ,niand M is minimally immersed iff H = 0.

The ChernConjectureBasicsWe also defineTheConjectureResultsS := |h| 2 = ∑ i,jh 2 ij = ∑ iλ 2 iGeneralizationsSummaryOutlookand for r ≥ 3f r := tr ((h ij ) r ) .withf 3 = ∑ h ij h jk h ki = ∑ λ 3 i , f 4 = ∑h ij h jk h kl h li = ∑i,j,kii,j,k,l iλ 4 i .

ExampleThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlook

The ConjectureThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe Chern conjecture for isoparametric hypersurfaces inspheres can be stated as follows:Let M be a closed, minimally immersed hypersurface of the(n + 1)-dimensional sphere S n+1 with constant scalarcurvature. Then M is isoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookIt was originally proposed in a less strong version by Chern(1968) and in Chern, do Carmo and Kobayashi (1970), andfinally by Verstraelen (1986) in the version presented above.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlook

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookSo far, no proof for the conjecture has been found, althoughpartial results exist in particular for low dimensions (especiallyup to four) and with additional conditions for the curvaturefunctions on M.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookIts original version relates to the following theorem, first provedby Simons (1968):TheoremLet M ⊂ S n+1 be a closed, minimally immersed hypersurfaceand S the squared norm of its second fundamental form. Then∫(S − n)S ≥ 0.MIn particular, for S ≤ n one has either S = 0 or S = nidentically on M.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookIts original version relates to the following theorem, first provedby Simons (1968):TheoremLet M ⊂ S n+1 be a closed, minimally immersed hypersurfaceand S the squared norm of its second fundamental form. Then∫(S − n)S ≥ 0.MIn particular, for S ≤ n one has either S = 0 or S = nidentically on M.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookSince M is minimally immersed S is constant if and only if thescalar curvature κ is constant. In this case it follows that S = 0or S ≥ n, which led Chern to propose the following conjecture:Consider closed minimal hypersurfaces M ⊂ S n+1 withconstant scalar curvature κ. Then for each n the set of allpossible values for κ (or equivalently S) is discrete.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe only known examples for minimal hypersurfaces withconstant scalar curvature in S n+1 are isoparametric, i.e. all oftheir principal curvature functions are constant.One obtains that S equals (g − 1)n, where g is the number ofpairwise distinct principal curvatures and can only take thevalues 1, 2, 3, 4 or 6, which establishes the conjecture in thiscase.

ResultsThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe trivial case is given for n = 2:λ 1 + λ 2 = 0λ 2 1 + λ 2 2 = const

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe first partial result was achieved by Peng and Terng, whogave further constraints for the possible values of S:

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookTheorem (Peng, Terng 1983)For every n ≥ 3 there exists a maximal C(n) with the followingproperty: Let M ⊂ S n+1 be a closed minimal hypersurface withconstant S > n. Then it follows that S ≥ n + C(n) and onehas C(3) = 3, C(n) ≥ 112n .

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe lowest dimension for which the Chern conjecture isnon-trivial is n = 3. In this case, a more general theorem hasbeen proven:Theorem (Almeida, Brito 1990; Chang 1994)Let M ⊂ S 4 be a closed hypersurface with constant meancurvature H and constant scalar curvature κ. Then M isisoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookThe lowest dimension for which the Chern conjecture isnon-trivial is n = 3. In this case, a more general theorem hasbeen proven:Theorem (Almeida, Brito 1990; Chang 1994)Let M ⊂ S 4 be a closed hypersurface with constant meancurvature H and constant scalar curvature κ. Then M isisoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookAlmeida and Brito initially showed this under the additionalassumption that κ is non-negative. The approach of this proofhas since been used to show a number of other results. Changthen completed the proof. He proved this separately formanifolds with three everywhere distinct principal curvaturesand those where two principal curvatures coincide in a point, inthe former case generalizing a proof given earlier by Peng andTerng for minimal hypersurfaces.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookInstead of low dimensional manifolds, one can also considerthose with a certain number g of pairwise different principalcurvatures; g = 3 is the first non-trivial case, and one has thefollowing result:Theorem (Chang 1994)Let M ⊂ S n+1 be a closed hypersurface with constant meanand scalar curvatures which has exactly three pairwise distinctprincipal curvatures in every point. Then M is isoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookFor the case n = 4 a partial result has been proven under theadditional assumption that M is a Willmore hypersurface, i.e. acritical point of the Willmore functional W (M) := ∫ M ρn withρ 2 = S − nH 2 .

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookH. Li computed the Euler-Lagrange equation for the Willmorefunctional and obtained the following characterization:TheoremLet M n ⊂ S n+1 (1) be an n-dimensional compact hypersurface.Then M n is a Willmore hypersurface if and only if0 = −ρ n−2( 2HS − nH 3 − ∑ i,j,kh ij h jk h ki)+ (n − 1)∆(ρ n−2 H)− ∑ i,j(ρ n−2 ) ij (nHδ ij − h ij ),where ρ 2 = S − nH 2 , ∆ is the Laplacian and (.) ij is thecovariant derivative with respect to the induced connection.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookAn immediate consequence is the following characterization ofWillmore hypersurfaces in spheres with constant meancurvature and scalar curvature:TheoremLet M n ⊂ S n+1 (1) be an n-dimensional compact hypersurfacewith constant mean curvature and constant scalar curvature.Then M n is a Willmore hypersurface if and only iff 3 = ∑ i,j,kh ij h jk h ki = 2HS − 4H 3 .In particular the Willmore condition for minimal hypersurfaceswith constant scalar curvature is equivalent to the conditionf 3 ≡ 0.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookTheorem (Lusala, Scherfner, Sousa Jr. 2005)Let M ⊂ S 5 be a closed minimal Willmore hypersurface withconstant non-negative scalar curvature. Then M isisoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookTheorem (Scherfner, Sousa Jr. 2009)Let M ⊂ S 5 be a closed minimal Willmore hypersurface withconstant scalar curvature. Then M is isoparametric.

Bryant conjectureThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookOne obvious generalization is that on non-closed manifolds, i.e.a local version of the conjecture. This has in particular beenproposed by Bryant for the case n = 3:Let M ⊂ S 4 be a minimal hypersurface with constant scalarcurvature. Then M is isoparametric.

SummaryThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookn Chern Conjecture Chern Conjecture Chern Conjecture(H ≠ 0) (locally)2 Yes Yes Yes3 Yes Yes If S ≤ 3or g = 2 in p4 If f 3 ≡ 0, S ≤ 12 Yes (add. conditions) Yes (add. conditions)or f 3 const., S < 20 3or S < 37261or g ≡ 3> 4 If f 3 const., S

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryThis list is not complete, but exists...Outlook

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookFollowing Prof. S.-T. Yau we have the following situation:The general philosophy of the Chern conjecture is that if weassume a hypersurface in the sphere to be minimal, it is veryrigid. If we also make an extra assumption on the secondfundamental form, such as constant length, it will force themanifold to be more rigid. And we expect such constants to bediscrete and most likely to be finite. The relation withisoparametric hypersurfaces is interesting and perhaps typical.

OutlookThe ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookTheorem (Scherfner, Vrancken, Weiß 2009)Let M ⊂ S 7 be a closed hypersurface with H = f 3 = f 5 = 0,f 4 = const and κ ≥ 0. Then M is isoparametric.

The ChernConjectureBasicsTheConjectureResultsGeneralizationsSummaryOutlookMore ideas:4-dim. case for constant f 3 or KRelated questions for submanifolds of higher codimensionAnalyze heat kernels...