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Flexibility of Laplace-Beltrami eigenfunctions under metric variation

  • Andrei Agrachev
  • Nicola Da Ponte
  • Antonio Lerario
  • Matteo Testa
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Abstract

We study the surjectivity of the differential of the Laplace-Beltrami operator with respect to the metric parameter. We use the obtained results to prove that every small deformation of a nonconstant eigenfunction can be realized with a small perturbation of the metric. As a consequence, we prove that every generic property of smooth functions which is invariant under multiplication by a nonzero scalar is generic for eigenfunctions of the Laplace-Beltrami operator, extending a result by Uhlenbeck.

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Cite

BibTeXRIS
BibTeX
@misc{agrachev2026flexibility,
  title = {{Flexibility of Laplace-Beltrami eigenfunctions under metric variation}},
  author = {Andrei Agrachev and Nicola Da Ponte and Antonio Lerario and Matteo Testa},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03568},
  eprint = {2611.03568},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03568},
}
RIS
TY  - UNPB
TI  - Flexibility of Laplace-Beltrami eigenfunctions under metric variation
AU  - Andrei Agrachev
AU  - Nicola Da Ponte
AU  - Antonio Lerario
AU  - Matteo Testa
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03568
UR  - https://arxiv.org/abs/2611.03568
AB  - We study the surjectivity of the differential of the Laplace-Beltrami operator with respect to the metric parameter. We use the obtained results to prove that every small deformation of a nonconstant eigenfunction can be realized with a small perturbation of the metric. As a consequence, we prove that every generic property of smooth functions which is invariant under multiplication by a nonzero scalar is generic for eigenfunctions of the Laplace-Beltrami operator, extending a result by Uhlenbeck.
ER  -