Translating spacelike graphs by mean curvature flow with prescribed contact angle

Abstract

This article demonstrates the existence for all time of the nonparametric spacelike mean curvature flow with contact angle boundary condition, where the boundary manifold is a convex cylinder. We also consider the asymptotic behavior of the flow and prove that the flow converges to a spacelike hypersurface (unique up to translation) moving at a constant speed if solutions to an elliptic system exist.Mathematics Subject Classification. Primary 53C40, 53C44; Secondary 35K55.

On the Geometry of Spacelike Mean Curvature Flow Solitons Immersed in a GRW Spacetime

“…on some time interval, where H stands for the (nonnormalized) mean curvature vector of the spacelike submanifold Σ n t in R n+1 1 . Mean curvature flow in the Minkowski space and, more generally, in a Lorentzian manifold has been extensively studied by several authors (see, for example, [28,31,41]) and an important justification for this interest is the fact that spacelike self-shrinkers and, in a more general setting, spacelike mean curvature flow solitons (which constitute singularities of the spacelike mean curvature flow) can be regarded as a natural way of foliating spacetimes by almost null-like hypersurfaces. Particular examples may give insight into the structure of certain spacetimes at null infinity and have possible applications in general relativity.…”

Curvature estimates for spacelike graphic hypersurfaces in Lorentz–Minkowski space R1n+1$\mathbb {R}^{n+1}_{1}$

“…One might find that for curves and surfaces in $$\mathbb {R}^{3}_{1}$$, López's setting is more convenient than the one we have used here. Both settings have been used by us in previous works—see, for example, [12, 14] for the setting here and [10, 13] for López's. (3)In [9], Gao and Mao first considered the evolution of spacelike graphic hypersurface, defined over a convex piece of $$\mathcal {H}^{n}(1)$$ and contained in a time cone in $$\mathbb {R}^{n+1}_{1}$$ ($$n\ge 2$$), along the inverse mean curvature flow (IMCF for short) with zero Neumann boundary condition (NBC for short), and showed that this flow exists for all the time, the spacelike graphic property of the evolving hypersurfaces is preserved along flow, and after suitable rescaling, the rescaled hypersurfaces converge to a piece of the spacelike graph of a constant function defined over $$\mathcal {H}^{n}(1)$$ as time tends to infinity. Recently, the anisotropic versions of this conclusion (both in $$\mathbb {R}^{n+1}_{1}$$ and more general Lorentz manifold $$M^{n}\times \mathbb {R}$$) have been solved (see [10, 11]).…”

“…to deal with the PCPs in $$\mathbb {R}^{n+1}_{1}$$. Based on this reason, we prefer to go back to our treatment in [14] whose definitions for $$h_{ij}$$ and $$H$$ are the same with ones here. Through this philosophy, we use the setting $$\sigma _{n}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}$$ for the Gauss–Kronecker curvature in our study of IGCF with zero NBC in $$\mathbb {R}^{n+1}_{1}$$.…”

Gradient Estimates for Spacelike Mean Curvature Flow with Boundary Conditions

“…The perpendicular Neumann boundary condition was considered by the author in several settings [11][9] [8]. A recent article by G. Li, B. Gao and C. Wu [15] dealt exactly with the problem of general graphical angle conditions described below for general dimension n, however the key boundary gradient lemma in this paper is incorrect. Specifically equation (2.9) in that paper appears to come from differentiating the boundary condition in the normal direction into the domain where no such boundary condition holds.…”