Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

Year 2024, Volume: 17 Issue: 1, 157 - 170, 23.04.2024

Miguel Ortega , Buse Yalçın

https://doi.org/10.36890/iejg.1437356

Abstract

In this study, we deal with non-degenerate translators of the mean curvature flow in the well-known hyperbolic Einstein's static universe. We classify translators foliated by horospheres and rotationally invariant ones, both space-like and time-like. For space-like translators, we show a uniqueness theorem as well as a result to extend an isometry of the boundary of the domain to the whole translator, under simple conditions. As an application, we obtain a characterization of the the bowl when the boundary is a ball, and of certain translators foliated by horospheres whose boundary is a rectangle.

Keywords

Translator, mean curvature flow, hyperbolic Einstein, horospheres, rotationally invariant, hyperbolic space, elliptic PDEs

Ethical Statement

The authors declare that they have no interests/competing interests. No AI has been used to prepare this document. All pictures made by the authors of this paper with the aid of Maxima.

Supporting Institution

Miguel Ortega is partially financed by: (1) the Spanish MICINN and ERDF, project PID2020-116126GB-I00; (2) the “Maria de Maeztu” Excellence Unit IMAG, ref. CEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033. and (3) Research Group FQM-324 by the Junta de Andalucı́a. Buse Yalçın would like to thank the Erasmus grant from Ankara University and The Scientific and Techological Research Council of Türkiye (TÜBİTAK) Grant 2210-A.

Project Number

PID2020-116126GB-I00 and (TÜBİTAK) Grant 2210-A

Thanks

Buse Yalçın is grateful to the Institute of Mathematics (IMAG) and the Department of Geometry and Topology of Granada University for their hospitality.

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