Boundary Connected Sum of Escobar Manifolds
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Boundary Connected Sum of Escobar Manifolds Weiwei Ao1 · María del Mar González2 · Yannick Sire3 Received: 19 July 2018 © Mathematica Josephina, Inc. 2019
Abstract Let (X 1 , g¯ 1 ) and (X 2 , g¯ 2 ) be two compact Riemannian manifolds with boundary (M1 , g1 ) and (M2 , g2 ), respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum X = X 1 # X 2 by excising half ball near points on the boundary. The resulting metric on X has zero scalar curvature and a CMC boundary. We fully exploit the non-local aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very wellknown problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures. Keywords Connected sums · Escobar manifolds · Dirichlet-to-Neumann problems · Non-local techniques Mathematics Subject Classification 58A99 · 58J40
1 Introduction Let (X i , g¯i ), i = 1, 2, be two (n + 1)-dimensional smooth compact Riemannian manifolds with boundaries Mi , i = 1, 2, for some n ≥ 2, and set gi = g¯i | Mi . We
B
Yannick Sire [email protected] Weiwei Ao [email protected] María del Mar González [email protected]
1
Department of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
2
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
3
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
123
W. Ao et al.
assume that X i are scalar-flat in the interior and have constant mean curvature H0 on the boundary Mi . We are interested in constructing connected sums on the boundary, producing a new scalar-flat manifold with CMC of the boundary. This is related to the so-called Escobar problem (see [14]). Indeed, let u be a (positive) solution on X n+1 of the problem in X, − X u + n−1 4n R X u = 0 ∂ν u +
n−1 2 HM
u=
n+1 n−1 n−1 2 H0 u
on
∂ X,
where R X is the scalar curvature of X , HM the mean curvature of M = ∂ X , ν is the outer normal with respect to the metric g (on ∂ X ), and H0 is a constant depending only on the conformal structure. Therefore, the new metric 4
g¯ = u n−1 g¯ has zero curvature and the boundary has constant mean curvature with respect to the metric g¯ . Escobar in his seminal paper [14] solved the previous boundary problem in most of the cases (see also [5,19,20] for later developments on the problem). The aim of the present work is to provide a different construction based on connected sums. While connected sums in the interior have been obtained in [17,24], in this paper, we construct connected sums on the boundary. To this end, we fully use the analogy of the Escobar problem with the Dirichlet-to-Neumann operator approach as pointed out i
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