On scalar curvature rigidity of vacuum static spaces

Abstract

Abstract. In this paper we extend the local scalar curvature rigidity result in [7] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [11]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [17].

Volume comparison with respect to scalar curvature

“…If κ = 0, V -static metrics reduce to vacuum static metrics. Under same assumptions on g, Qing and the author showed that g is isometric to ḡ (see [14]). This rigidity result suggests that the borderline case κ = 0 is not necessary to be considered.…”

Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature

“…When restricted in conformal deformations, Qing and the author achieved a sharp rigidity result for vacuum static spaces with positive scalar curvature (c.f. [23]), which generalized Hang and Wang's work in [16] on upper hemisphere. On the other hand, motivated by a question proposed by Escobar ([14]), Barbosa, Mirandola and Vitorio found an elegant integral identity and with the aid of which they proved the rigidity part independently in a more general setting ( [2]).…”

“…Inspired by [5], Qing and the author generalized above rigidity result to generic vacuum static spaces ( [23]). This is in fact a sharp rigidity result due to Corvino's work on the stability of non-vacuum static domains ( [9]).…”

Critical metrics and curvature of metrics with unit volume or unit area of the boundary

“…There are several motivations to consider the weighted curvature functional F given in item (d) of Proposition 2.2, which was first introduced by A. E. Fischer and J. E. Mardsen in [21]. Indeed, this functional plays a fundamental role in the theory of deformation and rigidity of static and V-static manifolds as we can observe in the works by S. Brendle et al [12], S. Brendle and F. C. Marques [11], G. Cox, P. Miao and L.-F. Tam [15] and J. Qing and W. Yuan [39]. Proof.…”