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On stability of exponentially subelliptic harmonic maps

Xin Huang

TL;DR

The paper addresses the stability of exponentially subelliptic harmonic maps from compact sub-Riemannian manifolds to Riemannian manifolds by deriving the first and second variation formulas for the horizontal exponential energy $E(f)=\int_M e^{\frac{|df_H|^2}{2}}\,dv_g$. It shows that critical maps satisfy the Euler–Lagrange equation $\tau_H(f)=0$ and proves stability when the target has nonpositive curvature, while establishing instability for maps into spheres under explicit energy-density conditions. The results extend variational stability analysis to the subelliptic setting and provide concrete instability criteria for spherical targets. Overall, the work develops a rigorous variational framework for exponentially subelliptic harmonic maps and delineates when such maps are energetically favorable or prone to instability.

Abstract

In this paper, we study the stability problem of exponentially subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. We derive the rst and second variation formulas for exponentially subelliptic harmonic maps, and apply these formulas to prove that if the target manifold has nonpositive curvature, the exponentially subelliptic harmonic map is stable. Further, we obtain the instability of exponentially subelliptic harmonic maps when the target manifold is a sphere.

On stability of exponentially subelliptic harmonic maps

TL;DR

The paper addresses the stability of exponentially subelliptic harmonic maps from compact sub-Riemannian manifolds to Riemannian manifolds by deriving the first and second variation formulas for the horizontal exponential energy . It shows that critical maps satisfy the Euler–Lagrange equation and proves stability when the target has nonpositive curvature, while establishing instability for maps into spheres under explicit energy-density conditions. The results extend variational stability analysis to the subelliptic setting and provide concrete instability criteria for spherical targets. Overall, the work develops a rigorous variational framework for exponentially subelliptic harmonic maps and delineates when such maps are energetically favorable or prone to instability.

Abstract

In this paper, we study the stability problem of exponentially subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. We derive the rst and second variation formulas for exponentially subelliptic harmonic maps, and apply these formulas to prove that if the target manifold has nonpositive curvature, the exponentially subelliptic harmonic map is stable. Further, we obtain the instability of exponentially subelliptic harmonic maps when the target manifold is a sphere.
Paper Structure (3 sections, 6 theorems, 25 equations)

This paper contains 3 sections, 6 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The first and second variation formula
  3. Stability of exponentially subelliptic harmonic maps

Key Result

Lemma 2.1

If $f:(M, H, g_H) \to (N^n, h)$ is a smooth map, then for any $W \in f^{-1}TN$ we have where $\zeta = \pi_H(\nabla_{e_\alpha}e_\alpha)$ and $dv_g$ is the Riemannian volume element of $g$.

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • ...and 2 more