Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
Abstract
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+ nddcf, 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.
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BibTeX
@misc{fujihara2026convergence,
title = {{Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds}},
author = {Naotoshi Fujihara and Toru Kajigaya and Albert Wood},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23627},
eprint = {2609.23627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23627},
}
RIS
TY - UNPB
TI - Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
AU - Naotoshi Fujihara
AU - Toru Kajigaya
AU - Albert Wood
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23627
UR - https://arxiv.org/abs/2609.23627
AB - 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.
ER -