Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture
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- by Mohammad Ghomi and Ralph Howard PDF
- Proc. Amer. Math. Soc. 140 (2012), 4323-4335 Request permission
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.References
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Additional Information
- Mohammad Ghomi
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 687341
- Email: ghomi@math.gatech.edu
- Ralph Howard
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 88825
- Email: howard@math.sc.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4323-4335
- MSC (2010): Primary 53A05, 52A15; Secondary 37C10, 53C21
- DOI: https://doi.org/10.1090/S0002-9939-2012-11420-0
- MathSciNet review: 2957223