Problems related to isoparametric theory

Ge, Jianquan

doi:10.48550/arxiv.1910.12229

Cited by 2 publications

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References 52 publications

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“…Recall (cf. [4], [15]) that now M = M θ 0 = f −1 (c 0 ) is the minimal level hypersurface of the Cartan-Münzner isoparametric function f on S n+1 , where c 0 = m − −m + m + +m + = cos(gθ 0 ), 0 < θ 0 < π g , and f (p) = cos(gθ(p)) with θ(p) the distance of p ∈ S n+1 to M + (one of the two focal submanifolds M ± := f −1 (±1) with codimensions m ± + 1). Moreover, the parallel level sets M θ := f −1 (cos(gθ)), θ ∈ [0, π g ] (with M 0 = M + , M π g = M − ), constitute a singular Riemannian foliation of S n+1 .…”

Section: Integral Inequalities With Equalities By Ie Hypersurfacesmentioning

confidence: 99%

Integral-Einstein hypersurfaces in spheres

Ge,

2021

Preprint

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Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres, which is the main object of the Chern conjecture: such hypersurfaces are isoparametric. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus S 1 ( 1 n ) × S n−1 ( n−1 n ) and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with g ≤ 2 principal curvatures. In particular, all the minimal isoparametric hypersurfaces with g ≥ 3 principal curvatures are IE hypersurfaces. As applications, we also obtain some spherical Bernstein theorems.

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Section: Integral Inequalities With Equalities By Ie Hypersurfacesmentioning

confidence: 99%

Integral-Einstein hypersurfaces in spheres

Ge,

2021

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“…Its generalization to Finsler geometry is studied in [23,25,26,51,53]. More references can be found in the survey papers [19,35,46,50].…”

Section: Introductionmentioning

confidence: 99%

Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Xu¹

2020

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Let Mt be an isoparametric foliation on the unit sphere (S n−1 (1), g st ) with d principal curvatures. Using the spherical coordinates induced by Mt, we construct a Minkowski norm with the presentation F = r 2f (t), which generalizes the notions of (α, β)norm and (α 1 , α 2 )-norm. Using the technique of spherical local frame, we give an exact and explicit answer for the question when F = r 2f (t) really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry Φ between two Minkowski norms induced by Mt, which preserves the orientation and fixes the spherical ξ-coordinates. There are two ways to describe this Φ, either by a system of ODEs, or by its restriction to any normal plane for Mt, which is then reduced to a Hessian isometry between Minkowski norms on R 2 satisfying certain symmetry and d-properties. When d > 2, we prove this Φ can be obtained by gluing positive scalar multiplications and compositions between the Legendre transformation and positive scalar multiplications, so it must satisfy the (d)-property for any orthogonal decomposition R n = V ′ + V ′′ , i.e., for any nonzeroAs byproducts, we prove the following results. On the indicatrix (S F , g), where F is a Minkowski norm induced by Mt and g is the Hessian metric, the foliation Nt = S F ∩ R >0 M 0 is isoparametric. Laugwitz Conjecture is valid for a Minkowski norm F induced by Mt, i.e, if its Hessian metric g is flat on R n \{0} with n > 2, then F is Euclidean.

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Problems related to isoparametric theory

Abstract: In this note we briefly survey and propose some open problems related to isoparametric theory.

Cited by 2 publications

References 52 publications

Integral-Einstein hypersurfaces in spheres

Integral-Einstein hypersurfaces in spheres

Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

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