Regular Lagrangians are smooth Lagrangians

TL;DR

This work links the symplectic geometry of cotangent bundles with microlocal sheaf theory by studying the $\gamma$-completion of the space of smooth compact exact Lagrangians. It establishes that if the $\gamma$-support of a completed Lagrangian coincides with a smooth Lagrangian, then the completion is in fact that same Lagrangian, proving that regular Lagrangians are smooth. It further shows that a compact $\gamma$-support must be connected, ruling out fragmentation in the compact regime. The approach uses the Tamarkin category, sheaf quantization, and the notion of cohomologically chordless sheaves to transfer properties between Lagrangians and their sheaf-theoretic avatars, aided by contraction arguments for nearby-Lagrangian phenomena. Overall, the paper strengthens the bridge between spectral invariants, microlocal sheaf theory, and dynamics on cotangent bundles with potential implications for Hamilton–Jacobi theory and homogenization.

Abstract

We prove that for any element in the $γ$-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its $γ$-support is a smooth Lagrangian submanifold, then the element itself is a smooth Lagrangian. We also prove that if the $γ$-support of an element in the completion is compact, then it is connected.

Figures (2)

• Figure 6.1:
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Theorems & Definitions (42)

• Definition 1.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Definition 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof