On Rumin's Complex and Adiabatic Limits

TL;DR

This paper proves that harmonic forms on contact manifolds converge to Rumin's harmonic forms under a specific metric blow-up, confirming a conjecture and linking geometric limits to spectral sequence frameworks.

Contribution

It establishes the convergence of harmonic forms to Rumin's forms during metric degeneration, confirming a conjecture and connecting to spectral sequence approaches.

Findings

Harmonic forms converge to Rumin's forms under metric blow-up.

The result confirms Gromov's conjecture from 1994.

Spectral sequence reformulation of the convergence.

Abstract

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory spaces seen from within '', IHES preprint, 1994. This result can also be reformulated in terms of spectral sequences, after Forman, Mazzeo-Melrose. A key ingredient in the proof is the fact that the curvatures become unbounded in a controlled way.