Special Lagrangians in Milnor Fibers and Almost Lagrangian Mean Curvature Flow

Abstract

The focus of this thesis is twofold: (1) We solve the Shapere–Vafa Problem: We construct embedded special Lagrangian spheres in Milnor fibers. We give a necessary and sufficient condition for the existence of embedded special Lagrangian spheres in Milnor fibers. (2) We solve the Thomas–Yau Problem for Milnor fibers: We prove the Thomas–Yau conjecture for the almost Lagrangian mean curvature flow (ALMCF) for Milnor fibers, under the assumption that the initial Lagrangian is an embedded positive Lagrangian sphere satisfying a natural stability condition proposed by Thomas–Yau but adapted to Milnor fibers by us. In addition, we formulate a new approach to resolving the Thomas–Yau conjecture in arbitrary almost Calabi–Yau manifolds. The Thomas–Yau conjecture proposes certain stability conditions on the initial Lagrangian under which the Lagrangian mean curvature flow (LMCF) exists for all time and converges to the unique special Lagrangian in the Hamiltonian isotopy class, and therefore also homology class of the initial Lagranigan. One of the reasons for studying LMCF in Calabi–Yau manifolds (or ALMCF in almost Calabi–Yau manifolds) is that the Lagrangian condition, as well as homotopy and homology classes, are preserved. Therefore, if the flow converges, it converges to a special Lagrangian. We develop a method for finding special Lagrangian spheres in Milnor fibers. We provide examples which illustrate different situations which occur (the total number of special Lagrangian spheres is at least deg f − 1 and at most 1/2 deg f(deg f − 1), where f is the polynomial defining the Milnor fiber). We show that the almost Lagrangian mean curvature flow of Lagrangian spheres in Milnor fibers can be reduced to a generalized mean curvature flow of paths in C. This reduction is different from the one found by Thomas–Yau. We show that the limit of the flow is either a straight line segment or a polygonal line, corresponding to a special Lagrangian sphere or a chain of such spheres. We prove that under certain conditions (more general than the ones achieved by Thomas–Yau) the flow results in a special Lagrangian sphere. Finally, we develop a method for associating a curve in C with a compact Lagrangian in a more general setting of an almost Calabi–Yau manifold. We show that when the Lagrangian flows by ALMCF that the corresponding curve remains convex and shortens its length. The limit is either a straight line segment corresponding to a special Lagrangian or a polygonal line resulting in a decomposition of the original Lagrangian.