From Global to Local

From Global to Local

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Brendan Guilfoyle, Institute of Technology Tralee, Ireland
Fine Hall 314

In this talk we discuss aspects of the recent proof of the Caratheodory conjecture on the number of umbilic points on a closed convex surface in Euclidean 3-space. The proof starts with the global version of the conjecture for closed surfaces and ultimately leads to an index bound for the principal foliation of the surface about isolated umbilic points. This reverses the historic perspective on the conjecture and opens up a dichotomy between smooth and real analytic surfaces. The methods used include mean curvature flow with boundary, holomorphic discs and utilizes a Kaehler structure in which the metric is of indefinite signature (2,2).