https://neckpinch.com/authors/albert-wood/New arXiv papers by Albert Wood, 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/2401.02228arxiv:2401.02228Thu, 04 Jan 2024 00:00:00 GMTWei-Bo Su, Chung-Jun Tsai, Albert Wood — In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as t→∞. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas–Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family N^(ε(t)) constructed by gluing in special Lagrangian `Lawlor necks' of size ε(t), where the dynamics of the neck size ε(t) are driven by the obstruction for the existence of nearby special Lagrangians to N^(ε(t)). This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.Ricci flowMean curvature flowLagrangian mean curvature flowAncient solutionsSingularities https://arxiv.org/abs/2208.01574arxiv:2208.01574Tue, 02 Aug 2022 00:00:00 GMTJesse Madnick, Albert Wood — We study mean curvature flow of Lagrangians in ℂⁿ that are cohomogeneity-one with respect to a compact Lie group G ≤ SU(n) acting linearly on ℂⁿ. Each such Lagrangian necessarily lies in a level set μ⁻¹(ξ) of the standard moment map μcolon ℂⁿ → g^*, and mean curvature flow preserves this containment. We classify all cohomogeneity-one self-similarly shrinking, expanding and translating solutions to the flow, as well as cohomogeneity-one smooth special Lagrangians lying in μ⁻¹(0). Restricting to the case of almost-calibrated flows in the zero level set μ⁻¹(0), we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in μ⁻¹(0), we show it occurs as the Type II blowup model of a Lagrangian MCF singularity. Throughout, we give explicit examples of suitable group actions, including a complete list in the case of G simple. This yields infinitely many new examples of shrinking and expanding solitons for Lagrangian MCF, as well as infinitely many new singularity models.Mean curvature flowLagrangian mean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesHomogeneous spaces