Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Seunghyeok Kim, Monica Musso, Juncheng Wei

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Abstract

We concern C2-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the C2-compactness for all 4-manifolds (which may be non-umbilic). For the 5-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the C2-compactness for all 5-manifolds. Finally, we show that the C2-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

Original languageEnglish
Pages (from-to)1763-1793
Number of pages31
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume38
Issue number6
Early online date19 Feb 2021
DOIs
Publication statusPublished - 19 Oct 2021

Bibliographical note

Funding Information:
S. Kim is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2017R1C1B5076384, NRF2020R1C1C1A01010133, NRF2020R1A4A3079066), and the associate member problem of Korea institute for advanced study (KIAS). M. Musso has been supported by EPSRC research Grant EP/T008458/1. The research of J. Wei is partially supported by NSERC of Canada.

Keywords

  • Blow-up analysis
  • Boundary Yamabe problem
  • Compactness
  • Positive mass theorem

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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