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Mathematics > Differential Geometry

arXiv:0709.1644 (math)
[Submitted on 11 Sep 2007 (v1), last revised 17 Sep 2009 (this version, v3)]

Title:On the Area Functional of the Second Fundamental Form of Ovaloids

Authors:Steven Verpoort
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Abstract: The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.
Comments: 13 pages. (v2: As compared to a previous version of this article, which was entitled "The Mean Curvature of the Second Fundamental Form of an Ovaloid", there have been made substantial changes especially to the latter part of the article.) (v3: Reference to the work of M. Wiehe has been added.) Published version available here: this http URL
Subjects: Differential Geometry (math.DG)
MSC classes: 53A05, 53A07, 53A10, 53C45
Cite as: arXiv:0709.1644 [math.DG]
  (or arXiv:0709.1644v3 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.0709.1644
Journal reference: Mathematical Proceedings of the Royal Irish Academy 109A (2009) 2:187-200

Submission history

From: Steven Verpoort [view email]
[v1] Tue, 11 Sep 2007 14:58:55 UTC (8 KB)
[v2] Fri, 18 Jul 2008 21:21:51 UTC (13 KB)
[v3] Thu, 17 Sep 2009 20:57:47 UTC (13 KB)
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