Category Archives: Surfaces

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem. 1. Triangles in Hyperbolic Space Suppose we are given … Continue reading

Posted in Euclidean Geometry, Geometric structures, Hyperbolic geometry, Surfaces, Uncategorized | 1 Comment

Polygonal words

Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov: Question(Gromov): Let be a one-ended word-hyperbolic group. … Continue reading

Posted in Groups, Surfaces | Tagged , , , , , , , , | 6 Comments

Minimal laminations with leaves of different conformal types

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it … Continue reading

Posted in Complex analysis, Surfaces | Tagged , , , , , , | 8 Comments

Bridgeman’s orthospectrum identity

Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The -dimensional case is (in my opinion) the most beautiful, … Continue reading

Posted in Hyperbolic geometry, Special functions, Surfaces | Tagged , , , , | 5 Comments

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

Harmonic measure

An amenable group acting by homeomorphisms on a compact topological space preserves a probability measure on ; in fact, one can given a definition of amenability in such terms. For example, if is finite, it preserves an atomic measure supported … Continue reading

Posted in Dynamics, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , , | 1 Comment

The Goldman bracket

I was in Stony Brook last week, visiting Moira Chas and Dennis Sullivan, and have been away from blogging for a while; this week I plan to write a few posts about some of the things I discussed with Moira … Continue reading

Posted in Lie groups, Surfaces | Tagged , , , , , , | 5 Comments