On Hamiltonian stationarity of twisted Lagrangian tori in ${\mathbb{C}}^2$

TL;DR

The paper investigates when twisted Lagrangian tori $L_\gamma$ in $\mathbb{C}^2$ minimize area under Hamiltonian deformations. By explicitly parametrizing $L_\gamma$ via $F(\alpha,\beta)$ from $\gamma(\beta)=\rho(\beta)e^{i f(\beta)}$ and computing the induced metric and mean curvature, it shows that $L_\gamma$ is Hamiltonian stationary iff $\sqrt{\det g}\,C_{\rho,f}$ is constant, which forces $\gamma$ to be a circle about the origin. The main result is that $L_\gamma$ is Hamiltonian stationary if and only if $\gamma$ is a circle centered at $0$, implying Chekanov's exotic tori (arising from curves not enclosing the origin) are not area-minimizing in their Hamiltonian isotopy classes. This connects the base curve geometry to ambient Lagrangian geometry, resolves a question about area-minimization in the Hamiltonian setting for twisted tori, and ties into Oh's open question by showing product tori are the only stationary representatives in their class.

Abstract

Chekanov's exotic tori have been playing an important role in symplectic geometry as they are the only known examples of Lagrangian tori in ${\mathbb{C}}^2$ that are not Hamiltonian isotopic to a product torus. In this paper, we explore the differential geometry of a wider range of tori constructed by twisting simple closed planar curves, which include both certain product tori and Chekanov's exotic tori. In particular, we investigate the minimality of area of such twisted tori under Hamiltonian deformations and show that the only minimal twisted tori are the product ones. This tells us that Chekanov's exotic tori are not area minimal in their Hamiltonian isotopy classes.

Figures (2)

• Figure 1: The sketch on the left shows the annulus of inner radius $\rho_{min}$ and outer radius $\rho_{max}$ together with the two arcs joining the two boundary circles. The sketch on the right shows the lower half of the annulus deformed into a rectangle.
• Figure 2: The only way $\gamma$ does not have a self-intersection point is if it encloses the origin.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1