Some rigidity results for complete totally trapped submanifolds in generalized Robertson–Walker spacetimes

Abstract

In this work, we prove rigidity results for complete totally trapped spacelike submanifolds immersed in generalized Robertson–Walker spacetimes. In particular, we obtain uniqueness and non-existence results for totally trapped submanifolds. We use a maximum principle for the \(\infty \)-Laplacian in order to get our results. We also present examples of totally trapped submanifolds in the Schwarzschild black hole spacetime and a surface which is trapped but it is not totally trapped in the product spacetime \(-{\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {H}}^2\).