Spectral invariants for monotone Lagrangians

TL;DR

This paper extends spectral invariants to monotone Lagrangians within Floer theory, establishing their properties and demonstrating their use in studying symplectic rigidity of specific cases.

Contribution

It introduces spectral invariants for monotone Lagrangians, the broadest class with existing Floer theory, and proves their key properties for applications.

Findings

Spectral invariants are successfully extended to monotone Lagrangians.

The invariants satisfy crucial properties for applications.

Applications demonstrate symplectic rigidity results for certain monotone Lagrangians.

Abstract

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper "Symplectic topology as the geometry of generating functions," they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a "classical" Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.