https://neckpinch.com/authors/ryosuke-takahashi/New arXiv papers by Ryosuke Takahashi, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2002.05132/arxiv:2002.05132Wed, 12 Feb 2020 00:00:00 GMTRyosuke Takahashi — We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated (1,1)-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a C-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.Mean curvature flowLagrangian mean curvature flowLong-time existence & convergence https://neckpinch.com/papers/1912.13145/arxiv:1912.13145Tue, 31 Dec 2019 00:00:00 GMTRyosuke Takahashi — We study the line bundle mean curvature flow on Kähler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of Kähler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian-Yang-Mills equation away from a finite number of curves of negative self-intersection on the surface. As an application, we obtain a lower bound of a Kempf-Ness type functional on the space of potential functions satisfying the hypercritical phase condition.Mean curvature flowSingularities(Non)collapsingKähler geometryLong-time existence & convergence https://neckpinch.com/papers/1909.02452/arxiv:1909.02452Thu, 05 Sep 2019 00:00:00 GMTRyosuke Takahashi — We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the H-functional.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/1808.06997/arxiv:1808.06997Tue, 21 Aug 2018 00:00:00 GMTKeita Kunikawa, Ryosuke Takahashi — Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the "twistor energy" by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperkähler complex structure. In particular, our result implies some kind of energy gap theorem for hyperkähler manifolds which have no complex Lagrangian submanifolds.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence https://neckpinch.com/papers/1704.01879/arxiv:1704.01879Thu, 06 Apr 2017 00:00:00 GMTRyosuke Takahashi — We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.Ricci flowKähler–Ricci flowKähler geometryLong-time existence & convergence https://neckpinch.com/papers/1606.03019/arxiv:1606.03019Thu, 09 Jun 2016 00:00:00 GMTRyosuke Takahashi — On a polarized manifold (X,L), the Bergman iteration φₖ^((m)) is defined as a sequence of Bergman metrics on L with two integer parameters k, m. We study the relation between the Kähler-Ricci flow φₜ at any time t ≥ 0 and the limiting behavior of metrics φₖ^((m)) when m=m(k) and the ratio m/k approaches to t as k → ∞. Mainly, three settings are investigated: the case when L is a general polarization on a Calabi-Yau manifold X and the case when L=± K_X is the (anti-) canonical bundle. Recently, Berman showed that the convergence φₖ^((m)) → φₜ holds in the C⁰-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman's result and show that this convergence actually holds in the smooth topology.Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauKähler geometry