宇宙学常数不为零的静态度规的单调性公式
Monotonicity formulas for static metrics with non-zero cosmological constant
摘要：
本文研究了具有非零宇宙常数的非奇异真空静态时空。我们引入新的积分量，并在适当的假设下，我们证明了他们的单调性沿水平集流的静态潜力。然后，我们展示了如何使用这些属性，以获得一些尖锐的几何和分析不等式，其平等的情况下，可以用来描述旋转对称的基本静态解决方案。因此，我们能够证明一些新的唯一性声明的德西特和反德西特度量。特别是，我们证明了德西特解在具有连通边界和正宇宙学常数的三维静态度规中具有最小可能的表面重力。
英文摘要：
In this paper we study non-singular vacuum static space-times with non-zero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geometric and analytic inequalities, whose equality case can be used to characterize the rotational symmetry of the underlying static solutions. As a consequence, we are able to prove some new uniqueness statements for the de Sitter and the anti-de Sitter metrics. In particular, we show that the de Sitter solution has the least possible surface gravity among three-dimensional static metrics with connected boundary and positive cosmological constant.
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