Hyperk\"ahler Arnold Conjecture and its Generalizations

TL;DR

This paper extends the hyperk"ahler Arnold conjecture to multidimensional tori and degenerate cases, using Morse theory and finite-dimensional reduction, advancing the understanding of hyperk"ahler Floer theory.

Contribution

It generalizes and refines the hyperk"ahler Arnold conjecture, proving it for multidimensional tori and degenerate cases with new methods.

Findings

Proved the conjecture for multidimensional tori.

Established the degenerate version of the conjecture.

Developed a new approach using Morse theory and finite-dimensional reduction.

Abstract

We generalize and refine the hyperk\"ahler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperk\"ahler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley-Zehnder proof of the Arnold conjecture for the torus.