# Rigid fibers of spinning tops

## TL;DR

This paper identifies specific non-displaceable fibers in classical integrable systems like the Lagrangian and Kovalevskaya tops, using superheaviness, revealing unique non-displaceability properties.

## Contribution

It establishes the existence and uniqueness of non-displaceable fibers in a broad class of integrable systems, including classical tops, via superheaviness techniques.

## Key findings

- Identifies non-displaceable fibers in classical integrable systems.
- Shows the uniqueness of such fibers from the zero-section.
- Extends results to singular level sets of convex Hamiltonians.

## Abstract

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show that a singular level set of a convex Hamiltonian is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.