Complete translating solitons in Lorentzian products

TL;DR

This paper investigates the existence and properties of complete translating solitons in Lorentzian product manifolds, establishing conditions under which such solitons cannot exist due to curvature bounds and analyzing their geometric and analytic implications.

Contribution

It provides new obstructions to the existence of spacelike solitons based on curvature bounds and revises classical completeness results for spacelike submanifolds in Lorentzian products.

Findings

Curvature bounds influence soliton existence and completeness.

Affine and logarithmic bounds ensure completeness and the Omori-Yau principle.

Non-negative ambient Ricci curvature obstructs soliton existence.

Abstract

Obstructions to the existence of spacelike solitons depending on the growth of the mean curvature $H$ are proved for Lorentzian products $(M\times \mathbb{R}, \bar g=g_M-dt^2)$ with lowerly bounded curvature. The role of these bounds for both the completeness of the soliton $\Sigma$ and the applicability of the Omori-Yau principle for the drift Laplacian, is underlined. The differences between bounds in terms of the intrinsic distance $g$ of the soliton and the distance $g_M$ in the ambiance are analyzed, and lead to a revision of classic results on completeness for spacelike submanifolds. In particular, primary bounds, including affine $g$-bounds and logarithmic $g_M$-bounds, become enough to ensure both completeness and Omori-Yau's. Therefore, they become an obstruction to the existence of solitons when the ambiance Ricci is non-negative. These results, illustrated with a detailed example, deepen the uniqueness of solutions of elliptic equations which are not uniformly elliptic, providing insights into the interplay between mean curvature growth and the global properties of this geometric flow.