The Willmore energy and the magnitude of Euclidean domains
Copyright policy )The Willmore energy and the magnitude of Euclidean domains
We study the geometric significance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domain X X in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the function M X ( R ) = M a g ( R ⋅ X ) \mathcal {M}_X(R) = \mathrm {Mag}(R\cdot X) at R = ∞ R = \infty determines the Willmore energy of the boundary ∂ X \partial X . This disproves the Leinster-Willerton conjecture for a compact convex body in odd dimensions.
- University of Parma Italy
- University of Innsbruck Austria
- Lund University Sweden
Willmore energy, Mathematics - Differential Geometry, Metric Geometry (math.MG), Mathematics - Analysis of PDEs, Mathematics - Metric Geometry, Differential Geometry (math.DG), Mathematics - Classical Analysis and ODEs, Pseudodifferential and Fourier integral operators on manifolds, Euclidean domains, Classical Analysis and ODEs (math.CA), FOS: Mathematics, magnitude, Pseudodifferential operators, Other generalizations (nonlinear potential theory, etc.), Analysis of PDEs (math.AP)
Willmore energy, Mathematics - Differential Geometry, Metric Geometry (math.MG), Mathematics - Analysis of PDEs, Mathematics - Metric Geometry, Differential Geometry (math.DG), Mathematics - Classical Analysis and ODEs, Pseudodifferential and Fourier integral operators on manifolds, Euclidean domains, Classical Analysis and ODEs (math.CA), FOS: Mathematics, magnitude, Pseudodifferential operators, Other generalizations (nonlinear potential theory, etc.), Analysis of PDEs (math.AP)
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