Geodesics of positive Lagrangians in Milnor fibers

TL;DR

This paper investigates the existence and uniqueness of smooth geodesics among positive Lagrangian spheres in Milnor fibers, establishing a metric space structure on their Hamiltonian isotopy classes.

Contribution

It proves the existence and uniqueness of smooth geodesics for positive Lagrangians in Milnor fibers, providing new examples in arbitrary dimensions.

Findings

Existence of smooth geodesics in Milnor fibers

Uniqueness of solutions to geodesic equations

Induction of a metric space structure on Lagrangian classes

Abstract

The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric $\Upsilon$, which gives rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in $n$-dimensional $A_m$ Milnor fibers. We show the existence and uniqueness of smooth solutions to the initial value problem and the boundary value problem. In particular, we obtain examples of smooth geodesics of positive Lagrangians in arbitrary dimension. As an application, we show that the Riemannian metric $\Upsilon$ induces a metric space structure on the space of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in the above mentioned Milnor fibers.