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Naoyuki Koike

Publications and source records attributed to Naoyuki Koike.

At least 37 records · Page 2Linked to original sources
arXiv2018

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the $3$-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.

math.DG↗
arXiv2018

Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type

In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analytic, then it is a principal orbit of a Hermann type action on the symmetric space. A hyperpolar action on a symmetric space of non-compact type admits a reflective singular orbit if and only if it is a Hermann type action. Hence is not extra the assumption that the isoparametric submanifold admits a reflective focal submanifold. Also, we prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type satisfies some additional conditions, then it is a principal orbit of the isotropy action of the symmetric space, where we need not impose that the submanifold is of real analytic. We use the building theory in the proof.

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arXiv2017

Homogeneity of proper complex equifocal submanifolds

In this paper, we show that an irreducible proper complex equifocal submanifold of codimension greater than one in a symmetric space of non-compact type. The proof is performed by showing the homogeneity of the lift of the complexification of the original submanifold to some infinite dimensional anti-Kahelerian space.

math.DG↗
arXiv2017

Volume-preserving mean curvature flow for tubes in symmetric spaces

In this paper, we investigate the volume-prserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space. We prove that the tubeness is preserved along the flow under certain conditions.

math.DG↗
arXiv2017

Homogeneity of infinite dimensional anti-Kaehler isoparametric submanifolds II

In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is an orbit of the action of a Banach Lie group generated by one-parameter transformation groups induced by holomorphic Killing vector fields defined entirely on the ambient Hilbert space.

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arXiv2017

Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type (other than a (real) hyperbolic space) and the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature over the invariant submanifold in the $C^{\infty}$-topology in infinite time.

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arXiv2017

Gauss maps of the Ricci-mean curvature flow

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.

math.DG↗
arXiv2016

A modified mean curvature flow in Euclidean space and soap bubbles in symmetric spaces

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, where we note that the small spherical soap bubbles are invariant under the isotropy subgroup action of the isometry group of the symmetric space. Furthermore, we investigate the shape and the mean curvature of the small spherical soap bubbles.

math.DG↗
arXiv2015

Mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces

In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in $C^{\infty}$-topology.

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arXiv2015

The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action

In this paper, we study the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds, where "almost free" means that the stabilizers of the group action are finite. First we obtain the evolution equations for some geometric quantities along the regularized mean curvature flow. Next, by using the evolution equations, we prove a horizontally strongly convexity preservability theorem for the regularized mean curvature flow. From this theorem, we derive the strongly convexity preservability theorem for the mean curvature flow starting from compact Riemannian suborbifolds in the orbit space (which is a Riemannian orbifold) of the Hilbert Lie group action.

math.DG↗
arXiv2014

A holonomy invariant anisotropic surface energy in a Riemannian manifold

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is constant along each holonomy subbundle of the tangent bundle of the Riemannian manifold. First we obtain the first variational formula for this anisotropic surface energy. Next we shall introduce the notions of an anisotropic convex hypersurface, an anisotropic equifocal hypersurface and an anisotropic isoparametric hypersurface for this anisotropic surface energy. Also, we shall introduce the notion of an anisotropic tube for this anisotropic surface energy. We prove that anisotropic tubes over a one-point set in a symmetric space are anisotropic convex hypersurfaces and that anisotropic tubes over a certain kind of reflective submanifold in a symmetric space are anisotropic isoparametric and anisotropic equifocal hypersurfaces.

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arXiv2013

The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry

In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real analytic pseudo-Riemannian manifold. In particular, for a pseudo-Riemannian homogeneous space, we define another complexification and a (complete) anti-Kaehler metric on the complexification. One of main purposes of this paper is to find the interesting relation between these two complexifications (equipped with the anti-Kaehler metrics) of a pseudo-Riemannian homogeneous space. Another of main purposes of this paper is to show that almost all principal orbits of some isometric action on the first complexification (equipped with the anti-Kaehler metric) of a semi-simple pseudo-Riemannian symmetric space are curvature-adapted isoparametric submanifolds with flat section in the sense of this paper.

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arXiv2013

Collapse of the mean curvature flow for isoparametric submanifolds in non-compact symmetric spaces

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.

math.DG↗
arXiv2013

A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore, by using the Cartan type identity, we show that certain kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of non-compact type are principal orbits of Hermann actions.

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