Abstract
Project Number
PID2020-116126GB-I00 and (TÜBİTAK) Grant 2210-A
References
- [1] Alekseevsky, A. V., Alekseevsky, A. V.: Riemannian G-manifold with one-dimensional orbit space. Ann. Glob. Anal. Geom. 11, 197–211 (1993).
- [2] Altschuler, S. J., Wu, L. F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var 2 pp 101–111 (1994) https://doi.org/10.1007/BF01234317
- [3] Batista, M., de Lima, H. F.: Spacelike translating solitons in Lorentzian product spaces: nonexistence, Calabi-Bernstein type results and examples. Commun. Contemp. Math. 24 no. 8, 2150034, 20 pp (2022). https://doi.org/10.1142/S0219199721500346
- [4] Bueno, A.: Translating solitons of the mean curvature flow in the space H2 × R. J. Geom. 109 42, (2018). https://doi.org/10.1007/s00022-018- 0447-x
- [5] Bueno, A.: Uniqueness of the translating bowl in H2 × R. J. Geom. 111, 43 (2020). https://doi.org/10.1007/s00022-020-00555-2
- [6] Clutterbuck, J., Schnürer, O. C., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. 29, 281–293 (2007). https://doi.org/10.1007/s00526-006-0033-1
- [7] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. ISBN: 3-540-41160-7
- [8] Hoffman, D., Ilmanen, T., Martín, F., White, B.: Notes on translating solitons for mean curvature flow. In Minimal surface: integrable systems and visualisation. vol 349 of Springer Proc. Math. Stat.. pp 147-168. Springer, Cham, 2021
- [9] Kim, D.: Rotationally symmetric space-like translating solitons for the mean curvature flow in Minkowski space. J. Math. Anal. Appl. 488, Issue 2, (2020). 124086, doi: https://doi.org/10.1016/j.jmaa.2020.124086
- [10] Lawn, M.-A., Ortega, M.: Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity One Actions. Mediterr. J. Math. 19, 102 (2022). https://doi.org/10.1007/s00009-022-02020-7
- [11] de Lira, J. H., Martín, F.: Translating solitons in Riemannian products. J. Diff. Equations 266, Issue 12 (2019) 7780–7812. https://doi.org/10.1016/j.jde.2018.12.015
- [12] Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. 54, 2853–2882 (2015). https://doi.org/10.1007/s00526-015-0886-2
- [13] O’Neill, B.: Semi-Riemannian geometry, With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. New York, 1983.
- [14] Pipoli, G.: Invariant translators of the solvable group. Annali di Matematica 199, 1961–1978 (2020). https://doi.org/10.1007/s10231-020- 00951-0
- [15] Pipoli, G.: Invariant Translators of the Heisenberg Group. J Geom Anal 31, 5219–5258 (2021). https://doi.org/10.1007/s12220-020-00476-1
- [16] Ros, A., Sicbaldi, P.: Geometry and topology of some overdetermined elliptic problems. J. Differential Equations 255(2013), no.5, 951–977. https://doi.org/10.1016/j.jde.2013.04.027
- [17] Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Second edition. Texts in Applied Mathematics, 2. Springer- Verlag, New York, 2003. ISBN: 0-387-00177-8
- [18] wxMaxima, https://maxima.sourceforge.io/ Last accessed: 2023-July-13.
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.
References
- [1] Alekseevsky, A. V., Alekseevsky, A. V.: Riemannian G-manifold with one-dimensional orbit space. Ann. Glob. Anal. Geom. 11, 197–211 (1993).
- [2] Altschuler, S. J., Wu, L. F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var 2 pp 101–111 (1994) https://doi.org/10.1007/BF01234317
- [3] Batista, M., de Lima, H. F.: Spacelike translating solitons in Lorentzian product spaces: nonexistence, Calabi-Bernstein type results and examples. Commun. Contemp. Math. 24 no. 8, 2150034, 20 pp (2022). https://doi.org/10.1142/S0219199721500346
- [4] Bueno, A.: Translating solitons of the mean curvature flow in the space H2 × R. J. Geom. 109 42, (2018). https://doi.org/10.1007/s00022-018- 0447-x
- [5] Bueno, A.: Uniqueness of the translating bowl in H2 × R. J. Geom. 111, 43 (2020). https://doi.org/10.1007/s00022-020-00555-2
- [6] Clutterbuck, J., Schnürer, O. C., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. 29, 281–293 (2007). https://doi.org/10.1007/s00526-006-0033-1
- [7] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. ISBN: 3-540-41160-7
- [8] Hoffman, D., Ilmanen, T., Martín, F., White, B.: Notes on translating solitons for mean curvature flow. In Minimal surface: integrable systems and visualisation. vol 349 of Springer Proc. Math. Stat.. pp 147-168. Springer, Cham, 2021
- [9] Kim, D.: Rotationally symmetric space-like translating solitons for the mean curvature flow in Minkowski space. J. Math. Anal. Appl. 488, Issue 2, (2020). 124086, doi: https://doi.org/10.1016/j.jmaa.2020.124086
- [10] Lawn, M.-A., Ortega, M.: Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity One Actions. Mediterr. J. Math. 19, 102 (2022). https://doi.org/10.1007/s00009-022-02020-7
- [11] de Lira, J. H., Martín, F.: Translating solitons in Riemannian products. J. Diff. Equations 266, Issue 12 (2019) 7780–7812. https://doi.org/10.1016/j.jde.2018.12.015
- [12] Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. 54, 2853–2882 (2015). https://doi.org/10.1007/s00526-015-0886-2
- [13] O’Neill, B.: Semi-Riemannian geometry, With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. New York, 1983.
- [14] Pipoli, G.: Invariant translators of the solvable group. Annali di Matematica 199, 1961–1978 (2020). https://doi.org/10.1007/s10231-020- 00951-0
- [15] Pipoli, G.: Invariant Translators of the Heisenberg Group. J Geom Anal 31, 5219–5258 (2021). https://doi.org/10.1007/s12220-020-00476-1
- [16] Ros, A., Sicbaldi, P.: Geometry and topology of some overdetermined elliptic problems. J. Differential Equations 255(2013), no.5, 951–977. https://doi.org/10.1016/j.jde.2013.04.027
- [17] Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Second edition. Texts in Applied Mathematics, 2. Springer- Verlag, New York, 2003. ISBN: 0-387-00177-8
- [18] wxMaxima, https://maxima.sourceforge.io/ Last accessed: 2023-July-13.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebraic and Differential Geometry |
| Journal Section | Research Article |
| Authors | |
| Project Number | PID2020-116126GB-I00 and (TÜBİTAK) Grant 2210-A |
| Early Pub Date | April 6, 2024 |
| Publication Date | April 23, 2024 |
| Submission Date | February 15, 2024 |
| Acceptance Date | March 24, 2024 |
| Published in Issue | Year 2024 Volume: 17 Issue: 1 |
