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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture


Authors: Mohammad Ghomi and Ralph Howard
Journal: Proc. Amer. Math. Soc. 140 (2012), 4323-4335
MSC (2010): Primary 53A05, 52A15; Secondary 37C10, 53C21
Published electronically: March 30, 2012
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Abstract: We show that Carathéodory's conjecture on umbilical points of closed convex surfaces may be reformulated in terms of the existence of at least one umbilical point in the graphs of functions $ f\colon \mathbf {R}^2 \to \mathbf {R}$ whose gradient decays uniformly faster than $ 1/r$. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of $ f$ whose projections into $ \mathbf {R}^2$ are parallel to any given direction.


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Additional Information

Mohammad Ghomi
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: ghomi@math.gatech.edu

Ralph Howard
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: howard@math.sc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11420-0
PII: S 0002-9939(2012)11420-0
Keywords: Umbilical point, Carathéodory conjecture, Loewner conjecture, principal line, Möbius inversion, parallel surface, divergence theorem.
Received by editor(s): May 16, 2011
Published electronically: March 30, 2012
Additional Notes: The research of the first-named author was supported in part by NSF grant DMS-0806305.
Communicated by: Lei Ni
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.