Sheaves and symplectic geometry of cotangent bundles

TL;DR

This paper consolidates and simplifies existing microlocal sheaf theory results to explore symplectic geometry of cotangent bundles, recovering key theorems and proving new conjectures.

Contribution

It unifies previous preprints with simplified proofs and applies microlocal sheaf theory to derive fundamental symplectic geometry results and solve a three cusps conjecture.

Findings

Recovered Gromov nonsqueezing theorem

Established Gromov-Eliashberg rigidity

Proved a three cusps conjecture

Abstract

This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them to study the symplectic geometry of cotangent bundles. We explain how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, the existence of graph selectors, we prove a three cusps conjecture about curves on the sphere and we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.