Book 9B

Comparison, Convergence and Heat on Manifolds

Riemannian geometry, part two

Turning curvature bounds into control of volume and distance, collapsing and noncollapsing, limits of manifolds, structure theorems, and the heat equation on a closed manifold.

Main companionRiemannian Geometry · Peter Petersen
ReferenceComparison Theorems in Riemannian Geometry · Cheeger & Ebin

In preparation

1
Laplacian Comparison
The Riccati equation for the distance function, and the focusing of light.

In preparation
2
Volume Comparison
Bishop–Gromov, packing arguments, and why hyperbolic space fits trees.

In preparation
3
Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

In preparation
4
Convergence of Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

In preparation
5
Splitting and Soul Theorems
Lines force splittings, souls, and asymptotic cones.

In preparation
6
Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.

In preparation
7
The Heat Equation on a Manifold
Heat kernels, the maximum principle without boundary, Li–Yau, and the conjugate heat operator.

In preparation

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Comparison, Convergence and Heat on Manifolds

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