The Radon transform and higher regularity of surfaces minimising a Finsler area

Abstract: A Finsler metric is a smooth family of smooth norms on the tangent bundle of a manifold. One possible generalisation of the usual Riemannian notion of area in Finsler geometry is the Busemann-Hausdorff area functional. In this talk I will consider high-codimensional disk-type surfaces which minimise this area with respect to Plateau boundary conditions. $\\$ I will show that the Busemann-Hausdorff area functional fits into the Hildebrandt-von der Mosel framework on Cartan functionals. Existence of minimisers is then guaranteed under mild growth conditions of the Finsler metric. Higher regularity ($W^{2,2}_{\textrm{loc}} \cap C^{1,\mu}$) of minimisers can be achieved by using functional analytic properties of the Radon transform. $\\$ The latter is an operator which assigns a function on the $(n−1)$-sphere its mean by integration over $(m-1)$-dimensional subspheres. One crucial property of this operator is its equivariance with respect to a Lie group action on the sphere and the $m$-Grassmannian. An infinitesimal version of this equivariance yields the regularity results about area minimisers.

analysis of PDEsdifferential geometrymetric geometry

Audience: researchers in the topic

( slides | video )

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Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

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