On the spectral capacity of submanifolds

TL;DR

This paper investigates the spectral capacities of neighborhoods around various submanifolds in symplectic geometry, revealing conditions under which these capacities are zero or positive, and establishing new obstructions for Lagrangian embeddings.

Contribution

It introduces a quantitative Lagrangian control estimate linking spectral invariants, boundary depth, and holomorphic disk areas, and provides novel obstructions to Lagrangian embeddings.

Findings

Neighborhoods of nowhere coisotropic submanifolds have zero spectral capacity.

Neighborhoods of closed Lagrangian submanifolds have uniformly positive spectral capacity.

New obstructions to Lagrangian embeddings into symplectic balls are established.

Abstract

The infimum of the spectral capacities of neighbourhoods of a nowhere coisotropic submanifold is shown to be zero. In contrast, neighbourhoods of a closed Lagrangian submanifold, and of certain contact-type hypersurfaces, are shown to have uniformly positive spectral capacity. Along the way we prove a quantitative Lagrangian control estimate relating spectral invariants, boundary depth, and the minimal area of holomorphic disks. The Lagrangian control also provides novel obstructions to certain Lagrangian embeddings into a symplectic ball.