Tag Archives: minimal surface

Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the … Continue reading

Posted in Complex analysis, Euclidean Geometry, Surfaces | Tagged , , , , , , , | 4 Comments

Second variation formula for minimal surfaces

If is a smooth function on a manifold , and is a critical point of , recall that the Hessian is the quadratic form on (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of at … Continue reading

Posted in 3-manifolds, Surfaces | Tagged , , , , , , , | 9 Comments

The topological Cauchy-Schwarz inequality

I recently made the final edits to my paper “Positivity of the universal pairing in 3 dimensions”, written jointly with Mike Freedman and Kevin Walker, to appear in Jour. AMS. This paper is inspired by questions that arise in the … Continue reading

Posted in 3-manifolds, TQFT | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Groups with free subgroups (part 2)

In a previous post, I discussed some methods for showing that a given group contains a (nonabelian) free subgroup. The methods were analytic and/or dynamical, and phrased in terms of the existence (or nonexistence) of certain functions on or on … Continue reading

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