Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.
Wednesday, January 22, 2014
1:00 p.m.
BA 6183, 40 St George St.
Ph.D. Candidate: Daniel Mayost
Ph.D. Advisor: Robert McCann
Thesis Title: Applications of the signed distance function to surface geometry
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In studying the geometry of a submanifold, it is often convenient to represent the submanifold as the zero set of an appropriately chosen defi ning function. For a hypersurface, a natural function to consider is its signed distance function. In this thesis we study the di erential geometry of surfaces embedded in ${\bf R}^3$ by expressing the curvatures and principal directions of a surface in terms of the derivatives of its signed distance function. This allows us to derive many established and new results using simple multivariable calculus.
After first de fining the signed distance function to a surface and demonstrating its basic properties, we prove several integral formulas involving the surface’s principal curvatures, including a generalization of the divergence theorem. We then derive a complex linear di fferential equation in the principal radii and directions of the surface that is based on the Mainardi-Codazzi equations, and establish a connection between the function in this equation and the surface’s support function. These are used to obtain a characterization of the principal directions around an umbilical point on a surface, and to reduce the Loewner index conjecture for real analytic surfaces to a problem about the location of the roots of a specifi c class of polynomials. Finally, we extend our generalized divergence theorem to submanifolds of arbitrary co-dimension equipped with a Riemannian metric.
A copy of the thesis can be found here: Mayost_Thesis
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