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Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

Xuezhang Chen

Department of Mathematics & IMS, Nanjing University, Nanjing 210093, P. R. China

E-mail Address: xuezhangchen@nju.edu.cn

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and

Liming Sun

Department of Mathematics, Rutgers University, 110 Frenlinghuysen Road, Piscataway NJ 08854, USA

Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218, USA

E-mail Address: lsun@math.jhu.edu

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https://doi.org/10.1142/S0219199718500219Cited by:7 (Source: Crossref)

Abstract

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n \geq 3$ . We prove the existence of such conformal metrics in the cases of $n = 6, 7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$ , there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+ \infty$ .

Keywords:

AMSC: 53C21, 35J65, 58J05, 35J20

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