On the Negativity of Ricci Curvatures of Complete Conformal Metrics
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On the Negativity of Ricci Curvatures of Complete Conformal Metrics Qing Han1 · Weiming Shen2 Received: 21 March 2019 / Revised: 23 December 2019 / Accepted: 30 April 2020 © Peking University 2020
Abstract A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases. Keywords Negativity of Ricci curvatures · The singular Yamabe problem · Negative sectional curvatures Mathematics Subject Classification 53C21
1 Introduction Let (M, g) be a compact Riemannian manifold of dimension n without boundary, for n ≥ 3 , and Γ be a smooth submanifold in M. For (M, g) = (Sn , gSn ) , Loewner and Nirenberg [15] proved that there exists a complete conformal metric on Sn ⧵ Γ with a negative constant scalar curvature if and only if dim(Γ) > (n − 2)∕2 . Aviles The first author acknowledges the support of NSF Grant DMS-1404596. The second author acknowledges the support of NSFC Grant 11571019. * Weiming Shen [email protected] Qing Han [email protected] 1
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
2
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
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and McOwen [4] proved a similar result for the general manifold (M, g). As a consequence, we can take the dimension of the submanifold to be n − 1 and conclude the following result: In any compact Riemannian manifold with boundary, there exists a complete conformal metric with a negative constant scalar curvature. See [4]. For convenience, we always take the constant scalar curvature to be −n(n − 1) . In this paper, we will study whether Ricci curvatures of such a metric remain negative. For the case of positive scalar curvatures, the existence and asymptotic behaviors of solutions have been extensively studied over the years. We shall not discuss this case here, but refer to [5, 12, 19–21, 23]. There are several classical results for metrics with negative Ricci curvatures. Gao and Yau [7] proved that there exists a metric of negative Ricci curvature on every compact 3-dimensional manifold without boundary. Lohkamp [16] generalized this to arbitrary dimensions and proved that any manifold of dimension n ≥ 3 (compact or not) admits a complete metric of negative Ricci curvature. Restricted to conformal metrics, by solving det(Ric) = constant with a precise boundary asymptotics, Guan [8] and Gursky, Streets and Warren [9] proved that there exists a complete conformal metric with negative
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