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arXiv · 2109.10097

The Willmore energy and the magnitude of Euclidean domains

Heiko Gimperlein·Magnus Goffeng
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Abstract

We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the magnitude function at infinity determines the Willmore energy of the boundary. This disproves the Leinster-Willerton conjecture for a compact convex body in all odd dimensions.

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Heiko Gimperlein, Magnus Goffeng. 2021-09-21. The Willmore energy and the magnitude of Euclidean domains. https://doi.org/10.1090/proc%2F16163

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