2017/06/07 by Scott Parkins, Parkins, Scott, Glen Wheeler +1
Mathematics · #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C44
paper · pdf · doi:10.48550/arxiv.1706.02045
34 pages
arxiv created 2017/06/07 · arxiv updated 2017/06/08
We study the curve diffusion flow for closed curves immersed in the Minkowski plane M, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in M depending on its length. The indiactrix \partialU (where U⊂ℝ2 is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix I. This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the flow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged L2 sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has enclosed area equal to the enclosed area of the initial immersion.