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PreprintCC BY 4.0arXiv

A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds

  • Nan Wu
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Abstract

Let (Xn+1,g+) be a Poincaré--Einstein manifold with conformal infinity (Mn,[h]) of positive Yamabe constant Y(M,[h]). Then a relative comparison inequality Y1(X,M,[g̅])/(Y1(𝕊n+1+,𝕊n,[g𝕊+n+1])) ≥ (Y(M,[h])/Y(𝕊n,[g𝕊n]))n/(n+1) holds for the \textbf{type-I} Escobar--Yamabe compactification Y1(X,M,[g̅], with equality if and only if (X, g+) is isometric to the hyperbolic space (ℍn+1, gℍn+1). This confirms a conjecture raised by Sun-Yung A. Chang.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026sharp,
  title = {{A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds}},
  author = {Nan Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.13742},
  eprint = {2607.13742},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.13742},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds
AU  - Nan Wu
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2607.13742
UR  - https://arxiv.org/abs/2607.13742
AB  - Let (X^n+1,g_+) be a Poincaré--Einstein manifold with conformal infinity (M^n,[h]) of positive Yamabe constant Y(M,[h]). Then a relative comparison inequality Y_1(X,M,[g̅])/(Y_1(𝕊^n+1_+,𝕊^n,[g_𝕊_+^n+1])) ≥ (Y(M,[h])/Y(𝕊^n,[g_𝕊^n]))^n/(n+1) holds for the \textbf{type-I} Escobar--Yamabe compactification Y_1(X,M,[g̅], with equality if and only if (X, g_+) is isometric to the hyperbolic space (ℍ^n+1, g_ℍ^n+1). This confirms a conjecture raised by Sun-Yung A. Chang.
ER  -