Remarks on the geodesically completeness and the smoothing effect on asymptotically Minkowski spacetimes

TL;DR

This paper investigates the geometric and analytic properties of asymptotically Minkowski spacetimes, establishing geodesic completeness under certain conditions and proving the optimality of a key Sobolev estimate related to the Klein-Gordon operator.

Contribution

It demonstrates geodesic completeness of asymptotically Minkowski spacetimes under null non-trapping conditions and confirms the optimality of a Sobolev estimate for the Klein-Gordon operator.

Findings

Asymptotically Minkowski spacetimes are geodesically complete under null non-trapping.

The Sobolev index used in estimates is proven to be optimal.

Provides insights into the geometric and analytic structure of such spacetimes.

Abstract

In this note, we study a geometric property of asymptotically Minkowski spacetimes and an analytic property of the Klein-Gordon operator. Precisely, our first main results show that asymptotically Minkowski spacetimes are geodesically complete under a null non-trapping condition. Secondly, we prove that Sobolev index of a real principal type estimate used in the previous work is actually optimal.