×

The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary condition. (English) Zbl 1296.53131

Summary: This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially space-like. Using a blowdown argument, we show that under renormalisation this flow converges towards a homothetically expanding hyperbolic solution.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K59 Quasilinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Steven J. Altschuler and Lang F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations 2 (1994), no. 1, 101 – 111. · Zbl 0812.35063
[2] John A. Buckland, Mean curvature flow with free boundary on smooth hypersurfaces, J. Reine Angew. Math. 586 (2005), 71 – 90. · Zbl 1082.37043
[3] Klaus Ecker, Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Differential Geom. 46 (1997), no. 3, 481 – 498. · Zbl 0909.53045
[4] Klaus Ecker, Mean curvature flow of spacelike hypersurfaces near null initial data, Comm. Anal. Geom. 11 (2003), no. 2, 181 – 205. · Zbl 1129.53303
[5] Klaus Ecker and Gerhard Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), no. 3, 595 – 613. · Zbl 0721.53055
[6] Brendan Guilfoyle and Wilhelm Klingenberg. Proof of the Carathéodory conjecture. Ar\( \chi \)iv preprint, 2012. http://arxiv.org/abs/0808.0851.
[7] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237 – 266. · Zbl 0556.53001
[8] Gerhard Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations 77 (1989), no. 2, 369 – 378. · Zbl 0686.34013
[9] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. · Zbl 0884.35001
[10] Axel Stahl, Convergence of solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations 4 (1996), no. 5, 421 – 441. · Zbl 0896.35059
[11] Axel Stahl, Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations 4 (1996), no. 4, 385 – 407. · Zbl 0851.35053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.