https://neckpinch.com/authors/naoyuki-koike/New arXiv papers by Naoyuki Koike, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.21884arxiv:2610.21884Thu, 15 Oct 2026 00:00:00 GMTKurando Baba, Naoyuki Koike — Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.Mean curvature flowHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2607.25025arxiv:2607.25025Mon, 27 Jul 2026 00:00:00 GMTKurando Baba, Naoyuki Koike — 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.Mean curvature flowLong-time existence & convergence https://arxiv.org/abs/2503.01355arxiv:2503.01355Mon, 03 Mar 2025 00:00:00 GMTNaoyuki Koike, Sakura Nakaoka — 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.Mean curvature flowHomogeneous spaces https://arxiv.org/abs/2412.10711arxiv:2412.10711Sat, 14 Dec 2024 00:00:00 GMTNaotoshi Fujihara, Naoyuki Koike — In this paper, we study the graphical mean curvature flow in a warped product ᵣ G/K × 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 × 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.Mean curvature flowSingularitiesRotational symmetry & warped products https://arxiv.org/abs/2308.15790arxiv:2308.15790Wed, 30 Aug 2023 00:00:00 GMTTomoki Fujii, Naoyuki Koike — 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)× U(n)), Sp(n+1)/(Sp(1)× Sp(n)) or F₄/rm Spin(9), we classify the shapes of translators in G/K×ℝ given by the graphs of functions on G/K which are invariant under the isotropy action Kcurvearrowright G/K. Next, in the case where G/K is of higher rank, we investigate translators in G/K×ℝ given by the graphs of functions on G/K which are invariant under a hyperpolar action Hcurvearrowright G/K of cohomogeneity two.Mean curvature flowSelf-shrinkers & translatorsHomogeneous spaces