arXiv 2202.063462022-02math.DG

关于具有位势的次椭圆调和映射

On subelliptic harmonic maps with potential

Yuxin Dong, Han Luo, Weike Yu

备注：: 29 pages. Any comments are welcome! arXiv admin note: text overlap with arXiv:1903.04702

摘要：

设$(M,H,g_H;g)$是次黎曼流形，$(N,h)$是黎曼流形.对于光滑映射$u: M \to N$，我们考虑能量泛函$E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$，其中$\mathrm{d}u_H$是$u$的水平微分，$G:N\to \mathbb{R}$是$N$上的光滑函数. $E_G(u)$的临界映射称为具有势$G$的次椭圆调和映射。本文利用次椭圆热流研究了带位势的次椭圆调和映射的存在性问题。假设目标黎曼流形具有非正的截面曲率，位势$G$满足各种适当的条件，证明了当源流形是阶跃$2$次黎曼流形或阶跃$r$次黎曼流形（其次黎曼结构来自于一个紧黎曼叶理）时，Eells-Sampson型解的存在性.

英文摘要：

Let $(M,H,g_H;g)$ be a sub-Riemannian manifold and $(N,h)$ be a Riemannian manifold. For a smooth map $u: M \to N$, we consider the energy functional $E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$, where $\mathrm{d}u_H$ is the horizontal differential of $u$, $G:N\to \mathbb{R}$ is a smooth function on $N$. The critical maps of $E_G(u)$ are referred to as subelliptic harmonic maps with potential $G$. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has non-positive sectional curvature and the potential $G$ satisfies various suitable conditions, we prove some Eells-Sampson type existence results when the source manifold is either a step-$2$ sub-Riemannian manifold or a step-$r$ sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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