On the Lagrangian capacity of convex or concave toric domains

TL;DR

This paper computes the Lagrangian capacity for convex and concave toric domains, showing it equals the diagonal for 4-dimensional convex cases and extending this to all such domains under certain assumptions, confirming a conjecture.

Contribution

It establishes the Lagrangian capacity equals the diagonal for 4D convex toric domains and extends this to all convex or concave toric domains under a virtual perturbation assumption.

Findings

Lagrangian capacity of 4D convex toric domains equals the diagonal.

Extension of the result to all convex or concave toric domains under certain assumptions.

Confirmation of a conjecture for the Lagrangian capacity of ellipsoids.

Abstract

We establish computational results concerning the Lagrangian capacity, originally defined by Cieliebak-Mohnke. More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. Working under the assumption that there is a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology, we extend the previous result to any convex or concave toric domain. This result gives a positive answer to a conjecture of Cieliebak-Mohnke for the Lagrangian capacity of the ellipsoid.