Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture

Ghomi, Mohammad; Howard, Ralph

doi:10.1090/S0002-9939-2012-11420-0

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.

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