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Gradient Estimates for Spacelike Mean Curvature Flow with Boundary Conditions

Part of: Parabolic equations and systems Global differential geometry

Published online by Cambridge University Press:  29 November 2018

Ben Lambert
Ben Lambert*
Affiliation:
25 Gordon Street, London WC1H 0AY, UK (b.lambert@ucl.ac.uk)
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Abstract

We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

Keywords

mean curvature flowgradient estimatesNeumann boundary conditionssemi-Riemannian geometryquasi-linear PDEs

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics 35K59: Quasilinear parabolic equations 35K20: Initial-boundary value problems for second-order parabolic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 459 - 469
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Altschler, S. J. and Wu, L. F., Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differ. Equ. 2 (1994), 101–111.Google Scholar
2.Ecker, K., Interior estimates and longtime solutions for mean curvature flow of non-compact spacelike hypersurfaces in Minkowski space, J. Differ. Geom. 45 (1997), 481–498.Google Scholar
3.Ecker, K., Mean curvature flow of spacelike hypersurfaces near null initial data, Commun. Anal. Geom. 11 (2003), 181–205.Google Scholar
4.Ecker, K. and Huisken, G., Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547–569.Google Scholar
5.Ecker, K. and Huisken, G., Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Commun. Math. Phys. 135 (1991), 595–613.Google Scholar
6.Guan, B., Mean curvature motion of non-parametric hypersurfaces with contact angle condition, In Elliptic and parabolic methods in geometry (ed. Peters, A. K.), pp. 47–56 (Wellesley, MA, 1996).Google Scholar
7.Huisken, G., Non-parametric mean curvature evolution with boundary conditions, J. Differ. Equ. 77 (1989), 369–378.Google Scholar
8.Lambert, B., A note on the oblique derivative problem for graphical mean curvature flow in Minkowski space, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 82(1) (2012), 115–120.Google Scholar
9.Lambert, B., Construction of maximal hypersurfaces with boundary conditions, preprint (arxiv:1408.5309, 2014).Google Scholar
10.Lambert, B., The constant angle problem for mean curvature flow inside rotational tori, Math. Res. Lett. 21(3) (2014), 537–551.Google Scholar
11.Lambert, B., The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary condition, Trans. Amer. Math. Soc. 366 (2014), 3373–3388.Google Scholar
12.Li, G. and Salavessa, I. M. C., Mean curvature flow of spacelike graphs, Math. Z. 269 (3–4) (2011), 697–719.Google Scholar
13.Li, G., Gao, S. and Wu, C., Translating spacelike graphs by mean curvature flow with prescribed angle, Arch. Math. 103 (2014), 499–508.Google Scholar
14.Lieberman, G. M., Second order parabolic differential equations (World Scientific Publishing, 1996).Google Scholar
15.Lira, J. H. and Wanderly, G. A., Mean curvature flow of Killing graphs, Trans. Amer. Math. Soc. 367 (2015), 4703–4726.Google Scholar
16.Schnürer, O., Translating solutions to the second boundary value problem for curvature flows, Manuscripta Math. 108 (2002), 319–347.Google Scholar
17.Stahl, A., Convergence of solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differ. Equ. 4 (1996), 421–441.Google Scholar
18.Stahl, A., Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition, Cal. Var. Partial Differ. Equ. 4 (1996), 385–407.Google Scholar
19.Wheeler, V. M., Mean curvature flow of entire graphs in a half-space with a free boundary, J. Reine Angew. Math. 690 (2014), 115–131.Google Scholar
20.Wheeler, V. M., Non-parametric radially symmetric mean curvature flow with free boundary, Math. Z. 276(1) (2014), 281–298.Google Scholar