Lovelock引力理论的约束方程：一个新的$σ_k$ -Yamabe问题
The constraint equations of Lovelock gravity theories: a new $σ_k$-Yamabe problem
摘要：
本文研究Lovelock引力理论的约束方程。在一个空的，紧凑的，共形平坦的，时间对称的，和类空流形的情况下，我们证明了哈密顿约束方程成为一个概括的$σ_k$ -Yamabe问题。也就是说，流形的$σ_k$ -曲率的线性组合的规定。我们搜索解决方案的共形类的一个紧凑的流形。使用现有的结果$σ_k$ -Yamabe问题，我们描述了一些情况下，他们可以扩展到这个新的问题。这就需要研究一些多项式的可积性。我们以两种方式做到这一点：关于这个多项式的整个根的计算，这与多项式的代数性质有关;并且解析地寻求凹函数。这给出了几种存在共形解的情况。最后给出了负数量曲率流形上的隐函数定理，并在Lovelock理论接近于广义相对论的情况下得到了共形解.
英文摘要：
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 $σ_k$-Yamabe problem. That is to say, the prescription of a linear combination of the $σ_k$-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the $σ_k$-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.
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