A Remark on Soliton Equation of Mean Curvature Flow

2003/12/08 by L. Ma, Li Ma, Y. Yang +3
Mathematics · Physics and Astronomy · #53C42 #53C44 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG #msc:53C42 #msc:53C44

paper · pdf · doi:10.48550/arxiv.math/0312151

6 pages

arxiv created 2003/12/08 · openalex publication_date 2003/12/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note, we consider self-similar immersions F: ℝⁿ → ℝ^n+k of the Graphic Mean Curvature Flow of higher co-dimension. We show that the following is true: Let F(x) = (x,f(x)), x ∈ ℝⁿ be a graph solution to the soliton equation H(x) + F^\bot(x) = 0. Assume sup_ℝⁿ|Df(x)| ≤ C₀ < + ∞. Then there exists a unique smooth function f_∞: ℝⁿ→ ℝ^k such that f_∞(x) = lim_{λ→ ∞}f_λ(x) and f_∞(r x)=r f_∞(x) for any real number r\not= 0, where f_λ(x) = λ^-1f(λx).

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