指数次椭圆调和映射的不稳定性与Morse指标
Instability and Morse Index of Exponentially Subelliptic Harmonic Maps
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中文总结 AI 辅助
研究指数次椭圆调和映射的Morse指标，通过外在和内在两种测试空间建立有限性并给出估计，适用于球面、球面乘积、凸超曲面及Heisenberg幂零流形。
中文摘要 AI 辅助
设$M$为具有括号生成分布$H$的闭流形。我们研究指数次椭圆调和映射的Morse指标。在建立有限性之后，我们通过两个测试空间来估计该指标。外在测试空间将指标形式在投影环境平行标架上平均，从而将估计归结为涉及具有四次指数项的外在标量的迹恒等式。内在测试空间使用拉回丛的平行截面，在其上指标形式化为一个耦合二次型，当存在全局平行标架时该二次型是精确的。这些界适用于标准球面、球面的乘积以及凸超曲面。在闭Heisenberg幂零流形上，内在约化给出了指标的谱公式。
英文摘要
Let $M$ be a closed manifold with a bracket-generating distribution $H$. We study the Morse index of exponentially subelliptic harmonic maps. After establishing finiteness, we estimate the index through two test spaces. The extrinsic one averages the index form over projected ambient parallel frames, reducing the estimate to a trace identity involving an extrinsic scalar with a quartic exponential term. The intrinsic one uses parallel sections of the pullback bundle, on which the index form descends to a coupled quadratic form that is exact when a global parallel frame exists. The bounds apply to round spheres, products of spheres, and convex hypersurfaces. On a closed Heisenberg nilmanifold, the intrinsic reduction yields a spectral formula for the index.
