Weil-Petersson curves, traveling salesman theorems and minimal surfaces

Weil-Petersson curves, traveling salesman theorems and minimal surfaces

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Christopher Bishop, Stony Brook University

Zoom linkhttps://princeton.zoom.us/j/594605776

We describe several new characterizations of Weil-Petersson curves. These curves are the closure of smooth planar closed curves for the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Their work was motivated by problems in string theory, but the same class arises naturally in geometric function theory,  computer vision, and the theory of Schramm-Loewner evolutions (SLE). The new characterizations include quantities such as Sobolev smoothness, Mobius energy, fixed curves of biLipschitz involutions, Peter Jones's beta-numbers, the thickness of hyperbolic convex hulls, the total curvature of minimal surfaces in hyperbolic space, and the renormalized area of these surfaces. Moreover, these new characterizations extend to higher dimensions and remain equivalent there.  

A link to the lecture slides is available at http://www.math.stonybrook.edu/~bishop/lectures/