-
Recent Posts
- Roots, Schottky semigroups, and Bandt’s Conjecture
- Taut foliations and positive forms
- Explosions – now in glorious 2D!
- Dipoles and Pixie Dust
- Mapping class groups: the next generation
- Groups quasi-isometric to planes
- Div, grad, curl and all this
- A tale of two arithmetic lattices
- 3-manifolds everywhere
- kleinian, a tool for visualizing Kleinian groups
- Kähler manifolds and groups, part 2
- Kähler manifolds and groups, part 1
- Liouville illiouminated
- Scharlemann on Schoenflies
- You can solve the cube – with commutators!
- Chiral subsurface projection, asymmetric metrics and quasimorphisms
- Random groups contain surface subgroups
- wireframe, a tool for drawing surfaces
- Cube complexes, Reidemeister 3, zonohedra and the missing 8th region
- Orthocentricity
- Kenyon’s squarespirals
- Thurston talks on geometrization at Harvard
- Random turtles in the hyperbolic plane
- Surface subgroups of Sapir’s group
- Upper curvature bounds and CAT(K)
Blogroll
- 0xDE
- Area 777
- Combinatorics and more
- Deep street soul
- Evaluating E-Discovery
- floerhomology
- Gaddeswarup
- Geometric Group Theory
- Godel's lost letter and P=NP
- Images des mathematiques
- Jim Woodring
- Language Log
- Letters of note
- Low dimensional topology
- Math Overflow
- Mathematics under the microscope
- n-Category Cafe
- Noncommutative geometry
- Paul Krugman
- Persiflage
- Preposterous Universe
- Questionable content
- Quomodocumque
- Real Climate
- Scott McCloud
- Secret blogging seminar
- Sketches of topology
- T Calegari
- Tanya Khovanova
- Terry Tao
- Tim Gowers
- Tony Phillips
Books
Software
Recent Comments
Categories
- 3-manifolds (18)
- 4-manifolds (2)
- Algebraic Geometry (2)
- Biology (2)
- Commentary (4)
- Complex analysis (10)
- Convex geometry (2)
- Diophantine approximation (1)
- Dynamics (13)
- Ergodic Theory (8)
- Euclidean Geometry (8)
- Foliations (2)
- Geometric structures (5)
- Groups (31)
- Hyperbolic geometry (23)
- Knot theory (1)
- Lie groups (8)
- Number theory (2)
- Overview (2)
- Polyhedra (2)
- Probability (1)
- Projective geometry (1)
- Psychology (3)
- Riemannian geometry (1)
- Rigidity (2)
- Special functions (1)
- Surfaces (19)
- Symplectic geometry (3)
- TQFT (1)
- Uncategorized (5)
- Visualization (10)
Meta
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
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
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-manifolds, Besson-Courtois-Gallot, compression body, Dijkgraaf-Witten, Hamilton, hyperbolic manifolds, JSJ decomposition, Miles Simon, minimal surface, Perelman, Ricci flow, scalar curvature, TQFT, unitary, universal pairing, volume entropy, volume inequality
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
