Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive K\"ahler-Einstein manifolds

Abstract

We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive K\"ahler--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 K\"ahler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho = C\omega + ndd^cf$, 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 {\L}ojasiewicz--Simon inequality argument to upgrade to smooth convergence of the full flow.