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Long time behavior of leafwise heat flow for Riemannian foliations

Jesus A. Alvarez Lopez, Yuri A. Kordyukov

TL;DR

This work addresses the long-time behavior of leafwise heat flow on Riemannian foliations, showing that leafwise heat evolution preserves smoothness and converges to the leafwise harmonic projection $\Pi$ under bundle-like metrics. It develops a general criterion (Theorem $t:$general) for stability of $e^{-t\Delta}$ on $C^{\infty}$-sections and applies it to the leafwise de Rham complex with a natural bigrading, revealing leafwise Hodge decompositions and spectral-sequence implications. Consequences include a leafwise Hodge theory with coefficients, a deformation-retract result for the space of bundle-like metrics, infinite-dimensional leafwise harmonic spaces under certain leaf conditions, and a concrete description of the second term $E_2$ of the spectral sequence for Riemannian foliations, including dualities under orientability. These results connect foliation spectral data to geometric deformation theory and provide tools for studying adiabatic limits and tautness in foliations.

Abstract

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.

Long time behavior of leafwise heat flow for Riemannian foliations

TL;DR

This work addresses the long-time behavior of leafwise heat flow on Riemannian foliations, showing that leafwise heat evolution preserves smoothness and converges to the leafwise harmonic projection under bundle-like metrics. It develops a general criterion (Theorem general) for stability of on -sections and applies it to the leafwise de Rham complex with a natural bigrading, revealing leafwise Hodge decompositions and spectral-sequence implications. Consequences include a leafwise Hodge theory with coefficients, a deformation-retract result for the space of bundle-like metrics, infinite-dimensional leafwise harmonic spaces under certain leaf conditions, and a concrete description of the second term of the spectral sequence for Riemannian foliations, including dualities under orientability. These results connect foliation spectral data to geometric deformation theory and provide tools for studying adiabatic limits and tautness in foliations.

Abstract

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.

Paper Structure

This paper contains 7 sections, 25 theorems, 89 equations.

Table of Contents

  1. Introduction and main results
  2. The general result
  3. Case of the leafwise de Rham complex for Riemannian foliations
  4. Case of leafwise differential forms with appropriate coefficients
  5. The space of bundle-like metrics
  6. Dimension of the space of leafwise harmonic forms
  7. Application to the second term of the spectral sequence of Riemannian foliations

Key Result

Theorem 1.1

With the above notation, suppose that, on $C^\infty(E)$, there is a transversely elliptic first order differential operator $A$, and there are zero order differential operators $G$, $H$, $K$ and $L$ such that Then $\Pi$ defines a continuous operator on $C^\infty(E)$, we have the leafwise Hodge decomposition and $(t,\phi)\mapsto e^{-t{\Delta}}\phi$ defines a continuous map $[0,\infty]\times C^\in

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7: X. Masa Masa92; see also Alv1, Alv2
  • Proposition 2.1: J. Roe Roe87
  • Lemma 2.2
  • Lemma 2.3
  • ...and 22 more