Isoparametric hypersurfaces with four principal curvatures, IV

Chi, Quo-Shin

doi:10.4310/jdg/1589853626

Cited by 56 publications

(47 citation statements)

References 15 publications

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“…The global isoparametric hypersurfaces of M being an Euclidean space or a hyperbolic space are classified by E. Cartan [5]. For the case when M is a unit sphere, the classification was recently completed by Q. Chi [6] (see also the surveys [19], [22], and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Some geometric correspondences for homothetic navigation

Xu¹,

Matveev²,

Zhang³

2020

Publ. Math. Debrecen

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In this paper, we discuss the geodesic and Jacobi field correspondences for homothetic navigation, and then use them to find alternative proofs for some wellknown flag curvature and S-curvature formulas, and to fully answer the question when a homothetic navigation preserves the local symmetric property in the curvature sense. These geometric correspondences also help us find the local correspondence between isoparametric functions or isoparametric hypersurfaces before and after a homothetic navigation, which generalizes the classification works of Q. He et al. for isoparametric hypersurfaces in Randers space forms and Funk spaces.

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Section: Introductionmentioning

confidence: 99%

Some geometric correspondences for homothetic navigation

Xu¹,

Matveev²,

Zhang³

2020

Publ. Math. Debrecen

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“…Thus, as a result of our findings and thanks to the collective efforts leading to the final classification [9], the family of critical domains in the whole sphere can be completely classified, and is in fact much larger than the family of geodesic balls. Criticality for the heat content, in S n , implies in particular that any component of the boundary is a smooth algebraic variety, more precisely, it is the zero set of the restriction to S n of a harmonic polynomial in R n+1 satisfying precise algebraic conditions (Cartan-Müntzner polynomials).…”

Section: Introduction and Main Resultsmentioning

confidence: 63%

“…For general facts about isoparametric hypersurfaces, see for example [40]. The classification of isoparametric hypersurfaces in the sphere is a classical problem in differential geometry which started from Cartan in the 30's ( [7]) and which, after several important intermediate results, has been completed only recently ( [9]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the heat content functional and its critical domains

Savo

2021

Calc. Var.

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We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times $$t>0$$ t > 0 . We do that by first computing the first variation of the heat content, and then showing that $$\Omega $$ Ω is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established in [33] and [34]. The outcome is that $$\Omega $$ Ω is critical for the heat content at time t, for all $$t>0$$ t > 0 , if and only if $$\Omega $$ Ω admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments $$T_1(\Omega ),T_2(\Omega ),\dots $$ T 1 ( Ω ) , T 2 ( Ω ) , ⋯ , which generalize the torsional rigidity $$T_1$$ T 1 . We prove that $$\Omega $$ Ω is critical for all $$T_k$$ T k if and only if $$\Omega $$ Ω is critical for the heat content at every time t, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE’s theory; in some respects they generalize the properties of the foliation of $$\mathbf{R}^{n}$$ R n by Euclidean spheres.

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“…Those in Euclidean and hyperbolic spaces were classified in the 1930s [5,12,15]. For the most difficult case, those in a unit sphere, were recently completely solved [7]. It is a natural idea to generalize these theories to Finsler geometry.…”

Section: Introductionmentioning

confidence: 99%

Isoparametric hypersurfaces in Finsler space forms

Chen

Yin

et al. 2021

Sci. China Math.

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In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that in a conic Minkowski manifold, besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, which are all isoparametric hypersurfaces, there are probably other isoparametric hypersurfaces, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in kropina spaces with constant flag curvature.

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Isoparametric hypersurfaces with four principal curvatures, IV

Cited by 56 publications

References 15 publications

Some geometric correspondences for homothetic navigation

Some geometric correspondences for homothetic navigation

On the heat content functional and its critical domains

Isoparametric hypersurfaces in Finsler space forms

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