Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On critical maps of the horizontal energy functional between Riemannian foliations

Tian Chong, Yuxin Dong, Xin Huang, Hui Liu

TL;DR

The paper extends harmonic map theory to the setting of mappings between Riemannian foliations by introducing and analyzing the horizontal energy functional $E_{H,tilde{H}}(f)$ and its critical points, the horizontally harmonic maps. It develops the stress-energy framework and conservation laws to derive robust monotonicity formulas in both integrable (Euclidean-type) and nonintegrable (Heisenberg quotient $K_m$) horizontal geometries, as well as for transversally harmonic maps under curvature pinching. Leveraging these monotonicity results, the authors prove Jin-type Liouville theorems under natural asymptotic/decay conditions, providing rigidity results for horizontally and transversally harmonic maps in mixed conformally flat spaces and in curved foliated domains. The work thereby generalizes subelliptic and transverse harmonic map theories, offering new tools for transverse geometric analysis and potential applications to rigidity phenomena in foliation theory.

Abstract

In this paper, we consider critical points of the horizontal energy $E_{\HH}(f)$ for a smooth map $f$ between two Riemannian foliations. These critical points are referred to as horizontally harmonic maps. In particular, if the maps are foliated, they become transversally harmonic maps. By utilizing the stress-energy tensor, we establish some monotonicity formulas for horizontally harmonic maps from Euclidean spaces, the quotients $K_{m}$ of Heisenberg groups and also for transversally harmonic maps from Riemannian foliations with appropriate curvature pinching conditions. Finally, we give Jin-type theorems for either horizontally harmonic maps or transversally harmonic maps under some asymptotic conditions at infinity.

On critical maps of the horizontal energy functional between Riemannian foliations

TL;DR

The paper extends harmonic map theory to the setting of mappings between Riemannian foliations by introducing and analyzing the horizontal energy functional and its critical points, the horizontally harmonic maps. It develops the stress-energy framework and conservation laws to derive robust monotonicity formulas in both integrable (Euclidean-type) and nonintegrable (Heisenberg quotient ) horizontal geometries, as well as for transversally harmonic maps under curvature pinching. Leveraging these monotonicity results, the authors prove Jin-type Liouville theorems under natural asymptotic/decay conditions, providing rigidity results for horizontally and transversally harmonic maps in mixed conformally flat spaces and in curved foliated domains. The work thereby generalizes subelliptic and transverse harmonic map theories, offering new tools for transverse geometric analysis and potential applications to rigidity phenomena in foliation theory.

Abstract

In this paper, we consider critical points of the horizontal energy for a smooth map between two Riemannian foliations. These critical points are referred to as horizontally harmonic maps. In particular, if the maps are foliated, they become transversally harmonic maps. By utilizing the stress-energy tensor, we establish some monotonicity formulas for horizontally harmonic maps from Euclidean spaces, the quotients of Heisenberg groups and also for transversally harmonic maps from Riemannian foliations with appropriate curvature pinching conditions. Finally, we give Jin-type theorems for either horizontally harmonic maps or transversally harmonic maps under some asymptotic conditions at infinity.

Paper Structure

This paper contains 11 sections, 20 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Riemannian foliations
  3. Critical maps of the horizontal energy functional
  4. The stress-energy tensor and conservation laws
  5. Monotonicity formulas
  6. Mixed conformal Euclidean spaces
  7. The quotient space $K_m$ of the Heisenberg group
  8. Transversally harmonic maps under curvature conditions
  9. Jin-type results
  10. Mixed conformal Euclidean spaces
  11. Transversally harmonic maps

Key Result

Lemma 2.1

Let $(M, \mathcal{F} , g)$ be a Riemannian foliation with Levi-Civita connection $\nabla$. If $X, Y \in \mathfrak{X}(M)$ are basic, then so is $\pi_{\mathcal{H}}\left(\nabla_X Y\right)$. In fact, if $\bar{X}=d \pi_U(X)$ and $\bar{Y}=d \pi_U(Y)$ for the local Riemannian submersion $\pi_U$ : $(U, g) \

Theorems & Definitions (50)

  • Lemma 2.1: gromollMetricFoliationsCurvature2009a
  • Lemma 2.2: gromollMetricFoliationsCurvature2009a
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 3.1: cf. eellsSelectedTopicsHarmonic1983a
  • Proposition 3.2
  • proof
  • ...and 40 more