Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Abstract

Abstract. We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions n = 3, 4, 5. As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique. Statement of the results.Let (M, g) be a smooth closed Riemannian manifold of dimension n ≥ 3. We are interested in the Einstein-scalar field Lichnerowicz equation in M :(EL) △ g u + hu = f u 2 * −1 + a u 2 * +1 , where h, f and a are smooth functions on M ,n−2 is the critical Sobolev exponent for the Sobolev space H 1 , and we assume that △ g + h is coercive, a ≥ 0, a ≡ 0, and max M f > 0. By closed, following standard terminology, we mean compact without boundary. Equation (EL) arises in the mathematical analysis of general relativity when solving the Einstein equations in a scalar-field setting, when the gravity is coupled to a scalar-field ψ. Special important cases include the massive Klein-Gordon setting or the case of a positive cosmological constant Λ. Given a closed manifold (M, g) of dimension n ≥ 3 endowed with two smooth functions π and ψ and a (2, 0)-symmetric tensor field K, the Cauchy problem in general relativity consists in finding a Lorentzian manifold (M × R,g) together with a smooth functionψ on M × R such thatψ |M = ψ and ∂ nψ|M = π, where ∂ n denotes the normalized time derivative, such that K is the second fundamental form of the embedding M ⊂ M × R, and such that (M × R,g) satisfies the Einstein equations:where Ric(g) is the Ricci tensor ofg, R(g) is its scalar curvature and T is the stress-energy tensor-field. This tensor field depends ong, onψ and on some potential V , itself related toψ by some wave equation. As shown first by for the vacuum case, see also Choquet-Bruhat-Isenberg-Pollack [6], a necessary and sufficient condition for the existence of such ag on M × R is that the following system of equations in M is satisfied:where R(g) is the scalar curvature of g and ∇ refers to the Levi-Civita connection of g. By specifying some of the unknown initial data (g, K, ψ, π) and solving the system for the remaining data, the conformal method initiated by Lichnerowicz [17] allows to turn (1.1) into a system of 1 2 BRUNO PREMOSELLI elliptic partial differential equations of critical Sobolev growth, called the conformal constraint system of equations. For a survey reference on the constraint equations see and for further informations on the conformal method see Choquet-Bruhat, Isenberg and Pollack [7]. Essentially, the set of free data consists of (ψ, τ, π, U ), where ψ, τ, π are smooth functions in M and U is a smooth symmetric traceless and divergence-free (2, 0)-tensor in M . Given (ψ, τ, π, U ) an initial free data set, the conformal constraint system of equations, whose unknown...

Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

“…The first step is to prove an existence result for solutions to the Lichnerowicz equation. Very nice existence results for solutions to the Lichnerowicz equation are given in [18], [31,32] and [19]. We prove here an existence result suited to our applications.…”

“…In particular, the method of [10] cannot work any longer and the Lichnerowicz equation may admits multiple solutions, see e.g. [8], [31] and references therein.…”

Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3

“…Under a positive mass assumption on g + h, we prove that sequences of positive solutions to this equation converge in C 2 (M ), as θ → 0, either to zero or to a positive solution of the limiting equation g u + hu = f u 5 . We also prove that the minimizing solution of (1) constructed by the author in [15] converges uniformly to zero as θ → 0.…”

A pointwise finite-dimensional reduction method for a fully coupled system of Einstein–Lichnerowicz type

“…• We prove Theorem 1.1 assuming that the L ∞ norm of the coupling field X is small, depending on n, g, h, f, π, σ (see Section 2). This assumption is harmless, since smallness conditions on X are necessary for solutions of (1.1) to exist: see [26,37,38]. • Our choice of the coefficients (h, f, π, σ, X, Y ) is driven by the a priori stability analysis of [39].…”