https://neckpinch.com/authors/naotoshi-fujihara/New arXiv papers by Naotoshi Fujihara, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.23627arxiv:2609.23627Sun, 20 Sep 2026 00:00:00 GMTNaotoshi Fujihara, Toru Kajigaya, Albert Wood — We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies ρ= Cω+ ndd^cf, and which satisfy C>0. With analogous assumptions on the flow, we obtain subconvergence to an f-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that (M,g,f) is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.Mean curvature flowCurve shorteningLagrangian mean curvature flowEinstein metricsSingularitiesSurfacesKähler geometryHomogeneous spacesLong-time existence & convergenceKähler–Einstein & K-stability 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/2410.08960arxiv:2410.08960Fri, 11 Oct 2024 00:00:00 GMTNaotoshi Fujihara — The behavior of the curve shortening flow has been extensively studied. Gage, Hamilton, and Grayson proved that, under the curve shortening flow, an embedded closed curve in the Euclidean plane becomes convex after a finite time and then shrinks to a point while remaining convex. Moreover, Grayson extended these results to surfaces that are convex at infinity and proved results similar to those for plane curves. In this paper, we study the curve shortening flow on surfaces that are not convex at infinity. Specifically, we consider a warped product of a unit circle and an open interval with a strictly increasing warping function. In this setting, we can define a graph property for curves within these warped products. It is known that this graph property is preserved along the curve shortening flow. Similarly to the behavior of the curve shortening flow in the plane, we prove that the curve becomes a graph after a finite time under the curve shortening flow.Curve shorteningSingularitiesSurfacesRotational symmetry & warped products https://arxiv.org/abs/2309.16449arxiv:2309.16449Thu, 28 Sep 2023 00:00:00 GMTNaotoshi Fujihara — We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and ℝ. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.Mean curvature flowCurve shorteningLong-time existence & convergenceRotational symmetry & warped products https://arxiv.org/abs/2202.09956arxiv:2202.09956Mon, 21 Feb 2022 00:00:00 GMTNaotoshi Fujihara — In this paper, we study curve shortening flows on rotational surfaces in ℝ³. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.Curve shorteningLong-time existence & convergence