Book 5A

Complex Analysis and Conformal Geometry

An optional side track to uniformization

Holomorphic functions as conformal maps, Cauchy’s theorem, residues, harmonic functions, the Riemann mapping theorem, and uniformization as the two-dimensional ancestor of geometrization.

Main companionComplex Analysis · Stein & Shakarchi

In preparation

1
Holomorphic Functions Are Conformal
Complex derivatives as rotations and scalings, from airfoils to the Mercator map.

In preparation
2
Cauchy’s Theorem and Its Consequences
Cauchy’s integral formula, Liouville, and the maximum modulus principle.

In preparation
3
Residues and Fourier Transforms
Contour integration, the argument principle and Nyquist stability.

In preparation
4
Harmonic Functions and Conformal Mapping
The Dirichlet problem, Poisson’s kernel and the Riemann mapping theorem.

In preparation
5
Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

In preparation
6
Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.

In preparation

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Complex Analysis and Conformal Geometry

Contents