Convergence of Lagrangian mean curvature flow in Kähler–Einstein manifolds

The Torelli Theorem for $ALH^*$ Gravitational Instantons

“…Now, in general the Lagrangian mean curvature flow may develop a finite time singularity [22], which is expected to be related to the Harder-Narasimhan filtration of the Fukaya category [15]. However, using a quantitative version of the machinery of Li [17], the authors proved a quantitative local regularity theorem for the Lagrangian mean curvature flow in the present setting; see [4,Theorem 4.23] Theorem 2.6. [Theorem 4.23, [4]] Let X be a non-compact Calabi-Yau surface with Ricci-flat metric ω and holomorphic volume form Ω.…”

Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

“…In the following, we need pointwise estimates to show the convergence of GLMCF. Generalizing Li's computation in [17], we list some estimates. Results in Section 3.3 and 3.4 of [17] for LMCF in Kähler-Einstein manifolds remain true in our case with slight modifications by a function f ∈ C ∞ (M ).…”

“…In this direction, there are still few convergence results of LMCF in general Kähler-Einstein manifolds (including Ricci-positive case). For examples, Wang showed a convergence results for a graph [39], and Li showed some stability results of LMCF, that is, LMCF with exact mean curvature form converges to a minimal Lagrangian if the initial data is sufficiently close to a Hamiltonian-stable minimal Lagrangian [17].…”

“…Then we prove that the GLMCF exists for all time and converges exponentially fast to an f ω -minimal Lagrangian (Theorem 7.4). The main techniques we used in our proof are generalization of the ones developed by Li [17] in Kähler-Einstein manifolds.…”

ɛ 0-regularity for mean curvature flow from surface to flat Riemannian manifold

“…Smoczyk and Wang [8] proved long time existence and convergence results for Lagrangian mean curvature flow under some convexity assumptions. For other references, see [9][10][11][12], etc.…”