On the curvature and heat flow on Hamiltonian systems

TL;DR

This paper develops geometric analysis tools for Hamiltonian systems, including curvature, Laplacian, and heat flow, extending classical formulas and theorems to this setting.

Contribution

It introduces generalized Bochner--Weitzenb"ock and Laplacian comparison theorems for Hamiltonian systems, advancing the understanding of their geometric and analytic properties.

Findings

Generalized Bochner--Weitzenb"ock formula for Hamiltonian systems

Laplacian comparison theorem extension

Analysis of heat flow in Hamiltonian dynamics

Abstract

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenb\"ock formula and Laplacian comparison theorem, and study the heat flow.