论文标题
$α-$谐波图的存在和稳定性
Existence and Stability of $α-$ harmonic Maps
论文作者
论文摘要
在本文中，我们首先研究了$α-$ Energy功能，Euler-Lagrange操作员和$α$ - 压力张量。其次，结果表明，$α-$能量函数的临界点通过共形变形明确与谐波图明确相关。此外，$α-$谐波图是根据某些假设下的Riemannian歧管之间的任何平滑地图构建的。接下来，我们确定水平形成$α-$谐波图的纤维在下面的条件最小。然后，证明了从riemannian歧管到具有非阳性riemannian曲率的里曼歧管的任何$α-$谐波图的稳定性。最后，研究了从紧凑型歧管到标准单位球体的$α-$谐波图的不稳定性。
In this paper, we first study the $α-$energy functional, Euler-Lagrange operator and $α$-stress energy tensor. Second, it is shown that the critical points of $α-$ energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an $α-$harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal $α-$ harmonic maps are minimal submanifolds. Then, the stability of any $α-$harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of $α-$harmonic maps from a compact manifold to a standard unit sphere is investigated.
