On the curvature energy of Cartesian surfaces
type: Item
instance of: scholarly article
title: On the curvature energy of Cartesian surfaces
zbMATH Open document ID: 1479.49093
author: Domenico Mucci
published in: The Journal of Geometric Analysis
publication date: 2021-08-31T00:00:00Z
review text: The author studies (in codimension one) the lower semicontinuous envelope of Cartesian surfaces under the functional curvature functional defined by \textit{G. Anzellotti} et al. [Indiana Univ. Math. J. 39, No. 3, 617--669 (1990; Zbl 0718.49030)] which for a smooth surface \(M\subseteq\mathbb{R}^3\) can be defined as follows using the elementary symmetric curvatures \(k_1\) and \(k_2\): \[ \|M\|:=\mathcal{H}^2(M)+\int_M \sqrt{k_1^2+k_2^2}\,d \mathcal{H}^2 +\int_M |k_1k_2|\,d\mathcal{H}^2. \] To this aim, following the approach of [loc. cit.], he studies the class of currents that naturally arise as weak limits of Gauss graphs of smooth functions. The curvature measures are then studied in the non-parametric case. Concerning homogeneous functions, some model examples are studied in detail. Finally, a new gap phenomenon is observed.
reviewed by: Laurent Moonens
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7388795
zbMATH Keywords: curvature of surfaces
zbMATH Keywords: Cartesian currents
zbMATH Keywords: Gauss graphs
zbMATH Keywords: gap phenomenon
MaRDI profile type: Publication
full work available at URL: s12220-020-00601-0
OpenAlex ID: W3126633509
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DOI: 10.1007/S12220-020-00601-0
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description: scientific article; zbMATH DE number 7388795