On critical maps of the horizontal energy functional between Riemannian foliations

Tian Chong; Yuxin Dong; Xin Huang; Hui Liu

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arXiv · 2504.01498

On critical maps of the horizontal energy functional between Riemannian foliations

Tian Chong·Yuxin Dong·Xin Huang·Hui Liu

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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.

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Tian Chong, Yuxin Dong, Xin Huang, Hui Liu. 2025-04-02. On critical maps of the horizontal energy functional between Riemannian foliations. https://arxiv.org/abs/2504.01498

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