关于4-调和与ES-4-调和超球面的法向稳定性
On the normal stability of the 4-harmonic and the ES-4-harmonic hypersphere
AI总结：
本文研究小超球面作为4-调和与ES-4-调和映射的法向稳定性，证明其在两种变分问题下均不稳定且法向指标均为1。
摘要：
4-调和映射与ES-4-调和映射都是对已被充分研究的调和映射方程的两种高阶推广，后者由一个八阶非线性椭圆偏微分方程给出。由于涉及大量导数，要发现这两个变分问题在定性行为上的任何差异都非常困难。众所周知，小超球面 \\(\iota\colon\s^m(\frac{1}{2})\to\s^{m+1}\\) 既是4-能量的临界点，也是ES-4-能量的临界点，但迄今为止尚未研究二者在其稳定性方面是否存在差异。本文的主要贡献是证明：该小超球面无论作为4-调和超球面还是ES-4-调和超球面，在法向变分下均不稳定，并且在两种情形下其法向指标都等于1。
英文摘要：
Both 4-harmonic and ES-4-harmonic maps are two higher order generalizations of the well-studied harmonic map equation given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. It is well known that the small hypersphere \(ι\colon\s^m(\frac{1}{2})\to\s^{m+1}\) is a critical point of both the 4-energy as well as the ES-4-energy but up to now it has not been investigated if there is a difference concerning its stability. The main contribution of this article is to show that the small hypersphere is unstable with respect to normal variations both as 4-harmonic hypersphere as well as ES-4-harmonic hypersphere and that its normal index equals one in both cases.
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。
