Lorentz polarisation and isoperimetric inequality in Minkowski spacetime

Abstract

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.