The constraint equations of Lovelock gravity theories: a new $\sigma_k$-Yamabe problem

This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of an empty, compact, conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the $\sigma_k$-Yamabe problem. That is to say, the prescription of a linear combination of the $\sigma_k$-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the $\sigma_k$-Yamabe problem, we describe some cases in which they can be extended to this new problem. This requires to study the concavity of some polynomial. We do it in two ways: regarding the concavity of an entire root of this polynomial, which is connected to algebraic properties of the polynomial; and seeking analytically a concavifying function. This gives several cases in which a conformal solution exists. At last we show an implicit function theorem in the case of a manifold with negative scalar curvature, and find a conformal solution when the Lovelock theories are close to General Relativity.