Minkowski时空中的Lorentz极化与等周不等式
Lorentz polarisation and isoperimetric inequality in Minkowski spacetime
AI总结：
本文在Minkowski时空中引入Lorentz极化方法，证明了有限光锥依赖区域的等周不等式，揭示了类空超平面截断光锥在体积最大化中的最优性，并推广至双曲面和非时序超曲面的等周型不等式。
摘要：
在本文中，我们证明了维数大于或等于3的Minkowski时空中有限光锥的依赖区域的一个等周不等式。该不等式涉及两个量：依赖区域的体积，以及有限光锥的周长。它指出，在所有具有相同周长的有限光锥中，依赖区域的最大体积由被截断的类空超平面有限光锥实现。该等周不等式的一个新颖之处在于余维2的比较特征。我们引入Lorentz极化，通过研究相应的变分问题来证明该等周不等式。一个关键观察是，在Lorentz极化下，有限光锥的依赖区域具有单调性。我们证明了任何有限光锥都可以通过Lorentz极化进行变换，以近似一个具有相等或更小周长的被截断的类空超平面有限光锥。作为Lorentz极化方法的进一步应用，我们证明了以下等周型不等式：a) 对于Minkowski时空中双曲面内具有给定周长的集合，双曲面中的测地球具有该集合依赖区域的最大体积；b) 对于光锥（或双曲面）中带有边界的非时序超曲面，在边界周长固定的情况下，类空超平面圆盘具有最大面积。
英文摘要：
In this paper, we prove an isoperimetric inequality for the domain of dependence of a finite lightcone in the Minkowski spacetime of dimension greater than or equal to 3. The inequality involves two quantities: the volume of the domain of dependence, and the perimeter of the finite lightcone. It states that among all finite lightcones with the same perimeter, the maximal volume of the domain of dependence is achieved by the spacelike hyperplane truncated finite lightcone. A novelty of this isoperimetric inequality is the codimension 2 comparison feature. We introduce the Lorentz polarisation to prove the isoperimetric inequality by studying the corresponding variational problem. A key observation is the monotonicity of the domain of dependence of a finite lightcone under the Lorentz polarisation. We show that any finite lightcone can be transformed by Lorentz polarisations to approximate a spacelike hyperplane truncated finite lightcone with an equal or less perimeter. As further applications of the method of Lorentz polarisation, we prove the following isoperimetric type inequalities: a) For a set with the given perimeter in the hyperboloid in the Minkowski spacetime, the geodesic ball in the hyperboloid has the maximal volume of the domain of dependence of the set; b) For an achronal hypersurface with boundary in the lightcone (or the hyperboloid), given the perimeter of the boundary fixed, the spacelike hyperplane disk has the maximal area.
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