Geodesics of positive Lagrangians in Milnor fibers

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 ${\rm Y}$, 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 ${\rm Y}$ induces a metric space structure on the space of positive $O_n(\mathbb{R})$ invariant Lagrangian spheres in the above mentioned Milnor fibers.

arXiv comment38 pages; added references and background, improved exposition, fixed minor errors

References in corpus (7)

• The Space of Kähler metrics (II)
• BPS Structure of Argyres-Douglas Superconformal Theories
• The Calabi homomorphism, Lagrangian paths and special Lagrangians
• The degenerate special Lagrangian equation
• Ricci flow and the metric completion of the space of Kahler metrics
• Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics
• Curvature of the space of positive Lagrangians

Cited by in corpus (2)

• The degenerate special Lagrangian equation
• Geodesics of positive Lagrangians from special Lagrangians with boundary