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Geodesics of positive Lagrangians from special Lagrangians with boundary

Jake P. Solomon, Amitai M. Yuval

Abstract

Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE. We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder. Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.

Geodesics of positive Lagrangians from special Lagrangians with boundary

Abstract

Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE. We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder. Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.

Paper Structure

This paper contains 20 sections, 56 theorems, 381 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Statement of results
  4. Outline
  5. Acknowledgments
  6. Background
  7. Weinstein neighborhoods and immersed Lagrangian submanifolds with boundary
  8. Immersed submanifolds
  9. The space of immersed Lagrangian submanifolds
  10. Frechet manifold
  11. Calabi-Yau manifolds, special Lagrangians and the space of positive Lagrangians
  12. Lagrangians with cone points
  13. Oriented blowups and cone-smooth differential topology
  14. Cone-immersed Lagrangians
  15. Lagrangian and special Lagrangian cylinders
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(\Lambda_t)_{t\in [0,1]}$ be a geodesic of positive Lagrangians and let $(h_t)_{t\in[0,1]}$ denote the associated Hamiltonian. For $c \in \mathbb{R},$ let Then $L_c$ is a smooth immersed submanifold of $X \times [0,1]$ diffeomorphic to the cylindrical manifold $\left(h_0^{-1}(c)\setminus \mathop{\mathrm{Crit}}\nolimits((\Lambda_t)_t)\right)\times [0,1]$, and the map given by $\Phi_c(p,t) =

Figures (2)

  • Figure 1: The imaginary special Lagrangian cylinders corresponding to a geodesic $(\Lambda_t)_t$ according to Theorem \ref{['theorem: family']}.
  • Figure 2: The immersed manifold $K = [f: M \to N]$ satisfies the hypothesis of Lemma \ref{['lemma: embedded boundary point implies freeness']} and therefore is free. The restriction of $f$ to the component $C \subset M$ is injective near the point $q$ of the boundary component $B \subset \partial C,$ but away from $B$ it is two-to-one. The restriction of $f$ to $M\setminus C$ is given by $f|_C$ composed with a diffeomorphism $M \setminus C \to C.$ So $K$ has no embedded points. However, the point $p = [f|_B,q]$ is an embedded point of the boundary component $[f|_B: B \to N]$.

Theorems & Definitions (161)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 151 more