Instability and Morse Index of Exponentially Subelliptic Harmonic Maps

Abstract

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.

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BibTeX RIS

BibTeX

@misc{huang2026instability,
  title = {{Instability and Morse Index of Exponentially Subelliptic Harmonic Maps}},
  author = {Xin Huang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33789},
  eprint = {2609.33789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33789},
}

RIS

TY  - UNPB
TI  - Instability and Morse Index of Exponentially Subelliptic Harmonic Maps
AU  - Xin Huang
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33789
UR  - https://arxiv.org/abs/2609.33789
AB  - 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.
ER  -

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