Viterbo's spectral bound conjecture for homogeneous spaces

TL;DR

This paper proves Viterbo's spectral bound conjecture for compact homogeneous spaces, showing that certain Lagrangian submanifolds have uniformly bounded spectral distance to the zero section, extending to some immersed cases.

Contribution

It establishes the conjecture for homogeneous spaces and extends the result to some immersed Lagrangians considering Reeb chord length.

Findings

Spectral distance is uniformly bounded for Lagrangians in homogeneous spaces.

The result applies to some immersed Lagrangians with Reeb chord length considerations.

Supports Viterbo's conjecture in a new geometric setting.

Abstract

We prove a conjecture of Viterbo about the spectral distance on the space of compact exact Lagrangian submanifolds of a cotangent bundle $T^*M$ in the case where $M$ is a compact homogeneous space: if such a Lagrangian submanifold is contained in the unit ball bundle of $T^*M$, its spectral distance to the zero section is uniformly bounded. This also holds for some immersed Lagrangian submanifolds if we take into account the length of the maximal Reeb chord.