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PreprintarXiv licencearXiv

Local rigidity for the sharp Nash inequality

  • Liang Cheng
  • Zhenyu Lu
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Abstract

Let V be a nonempty relatively compact open subset of a smooth Riemannian manifold (Mn,g) without boundary. We prove that CN(V,g)≥ CN(ℝn) and ∫V Scg dμg≥0 imply Rmg=0 on V, where CN denotes the optimal constant in the Nash inequality and Scg denotes the scalar curvature. This improves an earlier rigidity result of Druet, Hebey, and Vaugon [Int. Math. Res. Not. (1999), no. 14, 735--779].

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026local,
  title = {{Local rigidity for the sharp Nash inequality}},
  author = {Liang Cheng and Zhenyu Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32560},
  eprint = {2610.32560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32560},
}
RIS
TY  - UNPB
TI  - Local rigidity for the sharp Nash inequality
AU  - Liang Cheng
AU  - Zhenyu Lu
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32560
UR  - https://arxiv.org/abs/2610.32560
AB  - Let V be a nonempty relatively compact open subset of a smooth Riemannian manifold (M^n,g) without boundary. We prove that C_N(V,g)≥ C_N(ℝ^n) and ∫_V Sc_g dμ_g≥0 imply Rm_g=0 on V, where C_N denotes the optimal constant in the Nash inequality and Sc_g denotes the scalar curvature. This improves an earlier rigidity result of Druet, Hebey, and Vaugon [Int. Math. Res. Not. (1999), no. 14, 735--779].
ER  -