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The sharp L2-Michael--Simon inequality

  • Jeffrey S. Case
  • Dawit Mengesha
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Abstract

Let n ≥ 3. We prove that if j : (Σn,g) → (ℝN,dx2) is an isometric immersion, then ∫Σ( lvert ∇ u rvert2 + (n(n−2))/4lvert H rvert2 u2 ) ≥ (n(n−2))/4Vol(Sn)2/n( ∫Σlvert u rvert2n/(n−2) )(n−2)/n for all u ∈ W1,2(Σ), and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael--Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive Chen--Fenchel--Willmore-type inequalities relating Lp-norms of the mean curvature vector to the volume of a compact submanifold of a simply-connected spaceform.

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Cite

BibTeXRIS
BibTeX
@misc{case2026sharp,
  title = {{The sharp $L^2$-Michael--Simon inequality}},
  author = {Jeffrey S. Case and Dawit Mengesha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12309},
  eprint = {2610.12309},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12309},
}
RIS
TY  - UNPB
TI  - The sharp L^2-Michael--Simon inequality
AU  - Jeffrey S. Case
AU  - Dawit Mengesha
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12309
UR  - https://arxiv.org/abs/2610.12309
AB  - Let n ≥ 3. We prove that if j : (Σ^n,g) → (ℝ^N,dx^2) is an isometric immersion, then ∫_Σ( lvert ∇ u rvert^2 + (n(n−2))/4lvert H rvert^2 u^2 ) ≥ (n(n−2))/4Vol(S^n)^2/n( ∫_Σlvert u rvert^2n/(n−2) )^(n−2)/n for all u ∈ W^1,2(Σ), and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael--Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive Chen--Fenchel--Willmore-type inequalities relating L^p-norms of the mean curvature vector to the volume of a compact submanifold of a simply-connected spaceform.
ER  -