Real and complex hedgehogs, their symplectic area, curvature and evolutes

Classical (real) hedgehogs can be regarded as the geometrical realizations of formal differences of convex bodies in $\mathbb{R}^{n+1}$. Like convex bodies, hedgehogs can be identified with their support functions. Adopting a projective viewpoint, we prove that any holomorphic function $h : \mathbb{C}^n \to \mathbb{C}$ can be regarded as the ‘support function’ of a complex hedgehog $\mathcal{H}_h$ in $\mathbb{C}^{n+1}$. In the same vein, we introduce the notion of evolute of such a hedgehog $\mathcal{H}_h$ in $\mathbb{C}^2$, and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a ‘Brunn–Minkowski theory for complex hedgehogs’ (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give first two results in this direction. We next return to real hedgehogs in $\mathbb{R}^{2n}$ endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on $\mathbb{R}^4$ endowed with a linear Kähler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider briefly the convolution of hedgehogs, and the particular case of hedgehogs in $\mathbb{R}^{4n}$ regarded as a hyperkähler vector space.

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Received 10 August 2019

Accepted 30 October 2020

Published 21 July 2021