Uniqueness and nonexistence of spacelike translating solitons in GRW spacetimes

Giovanni Molica Bisci; Henrique F. de Lima; Ary V.F. Leite; Marco A. L. Velásquez

doi:10.3934/dcdss.2024173

Article Contents

2025, Volume 18, Issue 8: 2097-2126. Doi: 10.3934/dcdss.2024173

This issue Previous Article On a second-order delay integro-differential equation in Banach spaces Next Article Instantaneous blow-up solutions for nonlinear Sobolev-type equations on the Heisenberg groups

Uniqueness and nonexistence of spacelike translating solitons in GRW spacetimes

1.
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy
2.
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil

^*Corresponding author: Giovanni Molica Bisci
^*Corresponding author: Giovanni Molica Bisci

Received: May 2024

Revised: September 2024

Early access: September 20, 2024

Published online: September 20, 2024

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study the uniqueness and nonexistence of $ n $-dimensional spacelike translating solitons of the mean curvature flow immersed in a generalized Robertson-Walker (briefly GRW) spacetime $ -I\times_fM^n $ obeying appropriated curvature constraints on the Riemannian fiber $ M^n $ which also involve the warping function $ f $, via a sharp Bochner type inequality jointly with several maximum principles related to a drift Laplacian naturally attached to such a spacelike translating soliton. In this setting, we also derive new Calabi-Bernstein type results concerning the mean curvature equation attached to entire spacelike translating graphs constructed over the Riemannian fiber of a GRW spacetime. Furthermore, applications to some standard models of GRW spacetimes, like the Einstein-de Sitter and steady state type spacetimes, are also presented.

Keywords:

Generalized Robertson-Walker spacetimes,
mean curvature flow,
spacelike translating solitons,
Bochner type inequality,
Calabi-Bernstein type results.

Mathematics Subject Classification: Primary: 53C42; Secondary: 53E10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	M. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. PDE, 25 (2006), 205-246. doi: 10.1007/s00526-005-0351-8.
[2]	M. R. Adames, Spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-Euclidean spaces, Comm. Anal. Geom., 22 (2014), 897-929. doi: 10.4310/CAG.2014.v22.n5.a6.
[3]	A. L. Albujer and L. J. Alías, Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Amer. Math. Soc., 137 (2009), 711-721. doi: 10.1090/S0002-9939-08-09546-4.
[4]	L. J. Alías, A. Caminha and F. Y. S. do Nascimento, A maximum principle at infinity with applications to geometric vector fields, J. Math. Anal. and Appl., 474 (2019), 242-247. doi: 10.1016/j.jmaa.2019.01.042.
[5]	L. J. Alías, A. Caminha and F. Y. S. do Nascimento, A maximum principle related to volume growth and applications, Ann. Mat. Pura Appl., 200 (2021), 1637-1650. doi: 10.1007/s10231-020-01051-9.
[6]	L. J. Alías and A. G. Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc., 143 (2007), 703-729. doi: 10.1017/S0305004107000576.
[7]	L. J. Alías, A. G. Colares and H. F. de Lima, On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, Bull. Brazilian Math. Soc., 44 (2013), 195-217.
[8]	L. J. Alías, J. H. de Lira and M. Rigoli, Mean curvature flow solitons in the presence of conformal vector fields, J. Geom. Anal., 30 (2020), 1466-1529. doi: 10.1007/s12220-019-00186-3.
[9]	L. J. Alías, J. H. de Lira and M. Rigoli, Stability of mean curvature flow solitons in warped product spaces, Rev. Mat. Complutense, 35 (2022), 287-309. doi: 10.1007/s13163-021-00394-y.
[10]	L. J. Alías, P. Mastrolia and M. Rigoli, Maximum principles and geometric applications, Springer Monographs in Mathematics. Springer, Cham, 2016. xvii+570 pp.
[11]	D. Bakry and M. Émery, Diffusions hypercontractives, In Seminaire de Probabilites XIX, 1983/84, Lecture Notes in Math., Springer, Berlin, 1123 (1985), 177-206.
[12]	M. Batista and H. F. de Lima, Spacelike translating solitons in Lorentzian product spaces: Nonexistence, Calabi-Bernstein type results and examples, Comm. Contemp. Math., 24 (2022), 2150034.
[13]	M. Batista, G. Molica Bisci, H. F. de Lima and W. F. Gomes, Solitons of the spacelike mean curvature flow in generalized Robertson-Walker spacetimes, New York J. Math., 29 (2023), 554-579,.
[14]	S. Bochner, Vector fields and Ricci curvature, Bull. Am. Math. Soc., 52 (1946), 776-797. doi: 10.1090/S0002-9904-1946-08647-4.
[15]	H. Bondi and T. Gold, On the generation of magnetism by fluid motion, Monthly Not. Roy. Astr. Soc., 110 (1950), 607-611.
[16]	Q. Chen and H. Qiu, Rigidity of Self-shrinkers and translating solitons of mean curvature flows, Adv. Math., 294 (2016), 517-531. doi: 10.1016/j.aim.2016.03.004.
[17]	G. Colombo, L. Mari and M. Rigoli, Remarks on mean curvature flow solitons in warped products, Disc. Cont. Dynam. Syst., 13 (2020), 1957-1991.
[18]	H. F. de Lima, W. F. Gomes, M. S. Santos and M. A. L. Velásquez, On the geometry of spacelike mean curvature flow solitons immersed in a GRW spacetime, J. Australian Math. Soc., 116 (2024), 221-256.
[19]	J. H. S. de Lira and F. Martín, Translating solitons in Riemannian products, J. Diff. Eq., 266 (2019), 7780-7812. doi: 10.1016/j.jde.2018.12.015.
[20]	Q. Ding, Entire spacelike translating solitons in Minkowski space, J. Funct. Anal., 265 (2013), 3133-3162. doi: 10.1016/j.jfa.2013.09.010.
[21]	K. Ecker, On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime, J. Austral. Math. Soc. Ser. A, 55 (1993), 41-59. doi: 10.1017/S1446788700031918.
[22]	K. Ecker, Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Diff. Geom., 46 (1997), 481-498.
[23]	K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., 135 (1991), 595-613. doi: 10.1007/BF02104123.
[24]	A. Freitas, H. F. de Lima, M. S. Santos and J. S. Sindeaux, Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime, Ann. Glob. Anal. Geom., 63 (2023), 35 pp.
[25]	B. Guilfoyle and W. Klingenberg, Mean curvature flow of compact spacelike submanifolds in higher codimension, Trans. Amer. Math. Soc., 372 (2019), 6263-6281. doi: 10.1090/tran/7766.
[26]	S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, vol. 1, Cambridge University Press, London, 1973.
[27]	F. Hoyle, A new model for the expanding universe, Monthly Not. Roy. Astr. Soc., 108 (1948), 372-382. doi: 10.1093/mnras/108.5.372.
[28]	G. Huisken and S. T. Yau, Definition of center of mass for isolated physical system and unique foliations by stable spheres with constant curvature, Invent. Math., 124 (1996), 281-311. doi: 10.1007/s002220050054.
[29]	H.-Y. Jian, Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Diff. Eq., 220 (2006), 147-162. doi: 10.1016/j.jde.2005.08.005.
[30]	B. Lambert and J. D. Lotay, Spacelike mean curvature flow, J. Geom. Anal., 31 (2021), 1291-1359. doi: 10.1007/s12220-019-00266-4.
[31]	S. Montiel, Uniqueness of spacelike hypersurface of constant mean curvature in foliated spacetimes, Math. Ann., 314 (1999), 529-553. doi: 10.1007/s002080050306.
[32]	H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19 (1967), 205-214. doi: 10.2969/jmsj/01920205.
[33]	B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
[34]	S. Pigola, M. Rigoli and A. G. Setti, Vanishing theorems on Riemannian manifolds and geometric applications, J. Funct. Anal., 229 (2005), 424-461, (2005). doi: 10.1016/j.jfa.2005.05.007.
[35]	A. Romero, R. M. Rubio and J. J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson-Walker spacetimes, Class. Quantum Gravity, 30 (2013), 1-13.
[36]	V. M. Xavier, Sobre Princípios do Máximo Relacionados ao Crescimento de Volume e suas Aplicações(in Portuguese), Ph.D. thesis, Universidade Federal do Ceará, Fortaleza, 2022.
[37]	R. Xu and T. Liu, Rigidity of complete spacelike translating solitons in pseudo-Euclidean space, J. Math. Anal. Appl., 477 (2019), 692-707. doi: 10.1016/j.jmaa.2019.04.057.
[38]	S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Selected works of Shing-Tung Yau. Part 1., 2 (1971), 101-128.
[39]	S.-T. Yau, Some Function-Theoretic Properties of Complete Riemannian Manifolds and their Applications to Geometry, Indiana Univ. Math. J., 25 (1976), 659-670. doi: 10.1512/iumj.1976.25.25051.