The symplectic Hadamard question

Dan Cristofaro-Gardiner

Abstract

We show that a simply connected symplectic manifold admitting a complete nonpositively curved compatible metric is symplectomorphic to R^{2n} with its standard symplectic form. This answers the "symplectic Hadamard question" of McDuff-Salamon in the affirmative.

Disclosure

“tial use case, sitting as it does at the intersection of symplectic, complex, and Riemannian ge- ometry. In this case, it took a back and forth over a few hours with ChatGPT 5.5 Pro to arrive at a proposed solution, which I then checked and refined over several days, also with the help of GPT. The paper is entirely written by”
PDF page 1

Classification: Substantial proof generation
Multiplier: 10
Verified: 2026-08-23T18:53:28.394315Z

Structural counts

Pages 6 pdf

Theorems 1 source

Lemmas 1 source

Propositions 2 source

Corollaries 0 source

Definitions 0 source

Displayed equations 21 source

Bibliography entries 13 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file hadamard_arxiv.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.