Entire downward translating solitons to the mean curvature flow in Minkowski space

2015/05/07 by Joel Spruck, Ling Xiao, Spruck, Joel +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1505.01581

openalex publication_date 2015/05/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we study entire translating solutions u(x) to a mean curvature flow equation in Minkowski space. We show that if Σ=\(x, u(x))| x∈ℝⁿ\ is a strictly spacelike hypersurface, then Σ reduces to a strictly convex rank k soliton in ℝ^{k, 1} (after splitting off trivial factors) whose "blowdown" converges to a multiple λ∈(0, 1) of a positively homogeneous degree one convex function in ℝ^k. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation.

Related

• Entire downward solitons to the scalar curvature flow in Minkowski space
• Complete translating solitons to the mean curvature flow in ℝ³ with nonnegative mean curvature
• Convexity and the Dirichlet problem of translating mean curvature flows
• Translating solitons of the mean curvature flow in the space\n \ℍ²\×\ℝ
• Mean curvature flow solitons in the presence of conformal vector fields