论文标题
在潜在的下谐图图上
On subelliptic harmonic maps with potential
论文作者
论文摘要
令$（m，h，g_h; g）$为次 - 里曼尼亚歧管，$（n，h）$是riemannian歧管。对于平滑地图$ u：m \至n $，我们考虑能量功能$ e_g（u）= \ frac {1} {2} {2} \ int_m [| \ mathrm {d} u_h |^2-2g（u）] $ g：n \ to \ mathbb {r} $是$ n $的平滑函数。 $ e_g（u）$的关键地图称为带有潜在$ g $的次要谐波图。在本文中，我们调查了通过亚椭圆形热流和电势的下细胞谐波图的存在问题。假设目标riemannian歧管具有非阳性的截面曲率，并且潜在的$ g $满足各种适当的条件，那么当源代码歧管是一个步骤-2 $ 2 $ $ 2 $ sub-riemannian歧管，或者$ r $ $ r $ $ r $ r $ r $ riemannian歧管时，我们证明了某些Eells-Sampson类型的存在，其sub-riemannian sub-riemannian结构来自Riemannian sub-Riemannian foriation a a in a in a themann foriation
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.
