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

Kim, Seunghyeok; Musso, Monica; Wei, Juncheng

doi:10.1016/j.anihpc.2021.01.005

Kim, Seunghyeok ¹ ; Musso, Monica ² ; Wei, Juncheng ³

¹ a Department of Mathematics and Research Institute for Natural Sciences, College of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea
² b Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
³ c Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1763-1793.

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Résumé

We concern $C^{2}$ -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 $C^{2}$ -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 $C^{2}$ -compactness for all 5-manifolds. Finally, we show that the $C^{2}$ -compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

DOI : 10.1016/j.anihpc.2021.01.005

Classification : 35B40, 35J65, 35R01, 53A30, 53C21
Mots-clés : Boundary Yamabe problem, Compactness, Blow-up analysis, Positive mass theorem

Affiliations des auteurs :

Kim, Seunghyeok ¹ ; Musso, Monica ² ; Wei, Juncheng ³

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     title = {Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Kim, Seunghyeok; Musso, Monica; Wei, Juncheng. Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1763-1793. doi : 10.1016/j.anihpc.2021.01.005. http://www.numdam.org/articles/10.1016/j.anihpc.2021.01.005/

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