arXiv subjects

Naoyuki Koike

Publications and source records attributed to Naoyuki Koike.

At least 19 recordsLinked to original sources

arXiv2026

Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows

We first derive a formula for the mean curvature and the squared norm of the shape operator of equifocal hypersurfaces in simply-connected irreducible symmetric spaces of compact type. The formulas are given explicitly in terms of the tangential focal data of the equifocal hypersurfaces. Third, we study the backward mean curvature flow for equifocal hypersurfaces. The long-time existence of this flow for an equifocal hypersurface was established by Liu and Radeschi. We analyze the time evolution of the mean curvature and the squared norm of the shape operator along the long-time solution, thereby we generalize the result of Liu and Terng for isoparametric hypersurfaces in the sphere. Our analysis also gives an extension of Liu-Terng conjecture on the backward mean curvature flows in the sphere to the simply-connected irreducible symmetric space of compact type.

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arXiv2025

A construction of curvature-adapted hypersurfaces in the product of symmetric spaces

In this paper, we give a construction of curvature-adapted hypersurfaces in the product $G_1/K_1\times G_2/K_2$ of (Riemannian) symmetric spaces $G_i/K_i$ ($i=1,2$). By this construction, we obtain many examples of curvature-adapted hypersurfaces in $G_1/K_1\times G_2/K_2$. Also, we calculate the eigenvalues of the shape operator and the normal Jacobi operator of the curvature-adapted hypersurfaces obtained by this construction.

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arXiv2025

Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces

In this paper, we illustrate the behaviour of the mean curvature flows starting from principal orbits of any commuting Hermann action of cohomogeneity two on irreducible rank two Riemannian symmetric spaces of compact type by using Mathematicae. In more detail, we illustrate the velocity vector fields of the curves (determined by the flows) on the orbit space (which is a 2-simplex) of the Hermann action by using Mathematcae. Also, we calculate the position of the point of the orbit space corresponding to the only minimal principal orbit of the Hermann action by using Mathematicae.

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arXiv2024

Isotropy invariant graphical mean curvature flows in warped products

In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial hypersurface is $K$-equivariant, then the $K$-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group $K$ acts naturally on both $G/K$ and $_r G/K \times I$. If the flow is graphical, then it follows from the $K$-equivariance of the flow that it can be described by using $K$-invariant functions on $G/K$. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that $G/K$ is a rank one symmetric space of compact type and the warping function $r$ satisfies certain additional properties. The proof is carried out by estimating the gradient of the $K$-invariant functions satisfying the flow equation.

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arXiv2024

Isoparametric submanifolds in a Riemannian Hilbert manifold

In this paper, we introduce the notion of a regularizable submanifold in a Riemannian Hilbert manifold. This submanifold is defined as a curvature-invariant submanifold such that its shape operators and its normal Jacobi operators are regularizable, where ``the operators are regularizable'' means that the operators are compact and that their regularized traces and the usual traces of their squares exist. Furthermore, we introduce the notion of an isoparametric submanifold in a Riemannian Hilbert manifold. This submanifold is defined as a regularizable submanifold with flat section and trivial normal holonomy group satisfying the constancy of the regularized mean curvatures in the radial direction of the parallel submanifolds. For a curvature-adapted regularizable submanifold $M$ with trivial normal holonomy group in a locally symmetric Riemannian Hilbert manifold, we prove that if, for any parallel normal vecrtor field $\widetildeξ$ of $M$, the shape operaors $A_{\widetildeξ_x}$ and the normal Jacobi operator $\widetilde R(\widetildeξ_x)$ are independent of the base point $x(\in M)$ (up to orthogonal equivalent), then it is isoparametric under some additional conditions. Also, we define the notion of an equifocal submanifold in a Riemannian Hilbert manifold. We prove that the principal orbits of a certain kind of Hilbert Lie group action on the Riemannian Hilbert manifold $\mathcal A_P^{H^s}$ consisting of all $H^s$-connections of a $G$-bundle $P$ over a compact Riemannian manifold $B$ are equifocal, where $G$ is a semi-simple Lie group and $s>\frac{1}{2}\,{\rm dim}\,B-1$.

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arXiv2023

Translators invariant under hyperpolar actions

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space $G/K$ of compact type which is invariant under a hyperpolar action on $G/K$. First, in the case of $G/K=SO(n+1)/SO(n)$, $SU(n+1)/S(U(1)\times U(n))$, $Sp(n+1)/(Sp(1)\times Sp(n))$ or $F_4/{\rm Spin}(9)$, we classify the shapes of translators in $G/K\times\mathbb R$ given by the graphs of functions on $G/K$ which are invariant under the isotropy action $K\curvearrowright G/K$. Next, in the case where $G/K$ is of higher rank, we investigate translators in $G/K\times\mathbb R$ given by the graphs of functions on $G/K$ which are invariant under a hyperpolar action $H\curvearrowright G/K$ of cohomogeneity two.

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arXiv2023

Invariant Calabi-Yau structures on punctured complexified symmetric spaces

In this paper, we show that $G$-invariant Calabi-Yau structures on the complexification $G^{\mathbb C}/K^{\mathbb C}$ of a symmetric space $G/K$ of compact type are constructed from solutions of a Monge-Amp$\grave{\rm e}$re type equation. Also, we give an explicit descriptions of the Monge-Amp$\grave{\rm e}$re type equation in the case where the rank of $G/K$ is equal to one or two. Furthermore, we prove the existence of solutions of the Monge-Amp$\grave{\rm e}$re type equation.

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arXiv2023

Calabi-Yau structures on the complexifications of rank two symmeric spaces

For a (Reimannian) symmetric space $G/K$ of compact type, the natural action of $G$ on its complexification $G^{\mathbb C}/K^{\mathbb C}$ (which is an anti-Kaehler symmetric space) is one of the isometric actions called ``Hermann type action''. Let $\psi$ be the $G$-invariant strictly plurisubharmonic $C^{\infty}$-function on an open set of $G^{\mathbb C}/K^{\mathbb C}$ arising from a $W$-invariant strictly convex $C^{\infty}$-function $\rho$ on an open set of a maximal abelian subspace $\mathfrak a^d$ of $\mathfrak p^d$, where $\mathfrak p^d$ is the subspace of the Lie algebra $\mathfrak g^d$ of $G^d$ such that $\mathfrak g^d=\mathfrak k\oplus\mathfrak p^d$ gives the Cartan decomposition associated to the dual symmetric space $G^d/K$ of $G/K$ and $W$ is the Weyl group associated to $\mathfrak a^d$. In this paper, we first give a new proof of a known relation between the complex Hessian of $\psi$ and the Hessian of $\rho$. This new proof is performed from the viewpoint of the orbit geometry of the Hermann type action $G\curvearrowright G^{\mathbb C}/K^{\mathbb C}$ and and the isotropy action $K\curvearrowright G^d/K$. Next we prove conceptionally that there exists a $C^{\infty}$-Calabi-Yau structure on the whole of the complexification $G^{\mathbb C}/K^{\mathbb C}$ in the case where $G/K$ is of rank two on the basis of this relation.

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arXiv2023

Isoparametric submanifolds in Hilbert spaces and holonomy maps

Let $π:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is a homothetic submersion of coefficient $a$, where $s$ is a non-negative integer, $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$. In particular, we prove that, if $s=0$, then ${\rm hol}_c$ has minimal regularizable fibres. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in $G$ by the holonomy map ${\rm hol}_c:\mathcal A_P^{H^0}\to G$ is an isoparametric submanifold in $\mathcal A_P^{H^0}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.

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arXiv2022

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

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arXiv2021

Equifocal submanifolds with non-flat section and topological Tits buildings

From the Lytchak's result for polar foliations on an irreducible simply connected symmetric space $G/K$ of compact type and rank greater than one, we can derive that there exists no equifocal submanifold with non-flat section whose codimension is greater than two in the symmetric space $G/K$. In the first-half part of this paper, we give a new proof of this non-existence theorem. The recipi of our new proof is as follows. Suppose that there exists an equifocal submanifold $M$ with non-flat section whose codimension is greater than two in an irreducible symmertric space $G/K$ of compact type and rank greater than one. We introduce the notion of a slice topology of $G/K$ associated to $M$. We consider the universal covering $π:\widehat{G/K}\to G/K$ of the slice topological space $G/K$ and give $\widehat{G/K}$ the manifold structure and the Riemannian metric such that $π$ is a Riemannian submersion onto the symmetric space $G/K$. First we show that a simplicial decomposition of the Riemannian manifold $\widehat{G/K}$ gives an irreducible topological Tits building of spherical type and rank greater than two. By applying Burns-Spatzier's theorem to this topological Tits building, we show that the Riemannian manifold $\widehat{G/K}$ is homothetic to the unit sphere. Furthermore, from this fact, we show that $G/K$ is isometric to a sphere, a complex projective space or a quaternionic projective space. This contradicts that $G/K$ is of rank greater than one. This is the recipi of our proof. In the second-half part, we estimate the codimension of $M$ from above by using the multiplicities of the roots of the root system of $G/K$. As its result, we can show that there exists no equifocal submanifold with non-flat section in some irreducible simply connected symmetric spaces of compact type.

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arXiv2021

On equifocal submanifolds with non-flat section in symmetric spaces of rank two

In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat section in other irreducible simply connected symmetric spaces of compact type and rank two.

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arXiv2020

Mean curvature flow for pinched submanifolds in rank one symmetric spaces

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the submanifold collapses to a round point in finite time, or it converges smoothly to a totally geodesic submanifold in infinite time. In this paper, we prove the similar results for the mean curvature flow starting from pinched closed submanifolds in (general) rank one symmetric spaces of compact type. Also, we prove that closed submanifolds in (general) rank one symmetric spaces of non-compact type collapse to a round point along the mean curvature flow under certain strict pinching condition for the norm of the second fundamental form.

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arXiv2020

The preservability of the curvature-adaptedness along the mean curvature flow

In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.

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arXiv2020

Gromov-Hausdorff-like distance function defined in the aspect of Riemannian submanifold theory

In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact $C^k$-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$. The most important fact in this discussion is as follows. The Hausdorff distance function between two spheres of mutually distinct radii isometrically embedded into the hypebolic space of curvature $c$ converges to zero as $c\to-\infty$. The key in the construction of the Gromov-Hausdorff-like distance function given in this paper is to define the distance of two $C^{k+1}$-isometric embeddings of distinct compact $C^k$-Riemannian manifolds into a higher dimensional Riemannian manifold by using the Hausdorff distance function in the tangent bundle of order $k+1$ equipped with the Sasaki metric. Furthermore, we show that the convergence of a sequence of compact Riemannian manifolds with respect to this distance function coincides with the convergence in the sense of R. S. Hamilton.

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arXiv2019

Calabi-Yau structure and special Lagrangian submanifold of the complexified symmeric space

It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riemannian metric corresponding to the Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi-Yau structure, we give constructions of special Lagrangian submanifolds of any phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.

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arXiv2018

Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space

In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow.

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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.

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