Abstract
We use an elliptic system of equations with complex coefficients for a set of complex-valued tensor fields as a tool to construct infinite-dimensional families of non-singular stationary black holes, real-valued Lorentzian solutions of the Einstein–Maxwell-dilaton-scalar fields-Yang–Mills–Higgs–Chern–Simons- equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.
Highlights
Einstein equations with negative cosmological constant by perturbation of known ones
We show that Lee’s theorem on existence of perturbed Poincare-Einstein Riemannian metrics [25, Theorem A] can be extended to complex valued “metrics”, and to more general equations, and that this can be used to construct stationary Lorentzian black hole solutions with large classes of matter sources
We will say that a complex valued symmetric tensor field g is a complex metric if g is symmetric and invertible
Summary
Complex-valued tensor fields over M are defined as sections of the usual (real) tensor bundles over M tensored with C. N , be a collection of complex valued fields, forming a section of a complex bundle over M. Let g be a complex metric and consider a collection of N equations of the form (2.1). Gij ∂i∂j ΦA = F A(g, ∂g, Φ, ∂Φ) , with some functions F A which will be assumed to depend smoothly upon their arguments This can be rewritten as the following collection of 2N equations for 2N real fields (RΦ, IΦ):. As we will be using an implicit function theorem around real valued Riemannian metrics, our perturbation of Rgij, as well as Igij will always be sufficiently small for the estimates and the isomorphism properties to remain valid
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