Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture

  title={Normal curvatures of asymptotically constant graphs and Carath{\'e}odory’s conjecture},
  author={Mohammad Ghomi and Ralph Howard},
We show that Caratheodory’s conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f : R → 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… 

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