MINI COURSE, TOPICS IN METRIC GEOMETRY

SCHEDULE

Nov 17. Lecture 1. 2:30-5:30 pm

Nov 18. Lecture 2. 2:30-5:30 pm

Nov 19. Lecture 3. 2:30-5:30 pm

Nov 20. Lecture 4. 2:30-5:30 pm

Nov 21. Lecture 5. 2:30-5:30 pm

Nov 24. Lecture 6. 4:00-6:30 pm

Nov 25. Lecture 7. 4:00-6:30 pm

NICE PROBLEMS IN ORTHODOX GEOMETRY.

TOPICS

Reshetnyak's Gluing Theorem and Semi-dispersing Billiards:
[BBI] = Burago--Burago--Ivanov, A Course in Metric Geometry

Theorem on convex embedding:
Alexandrov, Convex polyhedra.

Curvature almost 1 (Sphere theorems)

Berger--Klingenberg proof (was not covered in our course):
[CE] = Cheeger--Ebin Comparison geometry.

Gromov's proof of sphere theorem:
[G1] = Gromov, Sign and geometric meaning of curvature.
Eschenburg, Local convexity and nonnegative curvature...

Micallef--Moore's proof of sphere theorem:
[G1] + Moore, On stability of minimal spheres and a two-dimensional version of Synge's theorem.
for standard Synge's lemma see [CE]
Micallef--Moore, Minimal two-spheres and the topology...

Ricci-flow proof; I used mostly
Boehm--Wilking, Manifolds with positive curvature operators are space forms

Curvature almost 0

Almost flat manifolds. I gave a prrof which is similar to the proof in the following paper;
the proof is based on few theorems in "Curvature bounded below" (see below).
Kapovitch--Petrunin--Tuschmann, Nilpotency, Almost Nonnegative Curvature and the Gradient Push

Curvature almost -1

Simplicial volume and Mostow rigidity
Thurston, The Geometry and Topology of Three-Manifolds

Gromov--Thurston's example
Gromov--Thurston, Pinching constants for hyperbolic manifolds.

Curvature bounded above

Almost nonpositively curved 3-shpere.
Buser--Gromoll, On the almost negatively curved 3-sphere.

Example of Hass
Hass, Bounded 3-manifolds admit negatively curved metrics with concave boundary.

Isoperimetic inequlaities
Croke, A sharp four-dimensional isoperimetric inequality.
Kleiner, An isoperimetric comparison theorem.

Curvature bounded below

Yamaguchi fibration theorem:
Yamaguchi, Collapsing and Pinching Under a Lower Curvature Bound.

Bishop--Gromov inequality and Gromov's compactness theorem:
[G2] = Gromov M. Metric structures for Riemannian and non-Riemannian spaces.

Toponogov's comparison theorem; as in 3.5 of *Games with definitions*, see below.

Toponogov's splitting theorem.

Short basis and fundamental group.
Meyer, Toponogov's Theorem and Applications.

Alexandrov's spaces; curvature bounded below.
I mark by *Something* fragment of the book which we are writing (S. Alexander, V. Kapovitch and me).
The book is UNDER CONSTRUCTION; there are many mistakes and misprints there, but it is close
to the presentation in my lectures.

*Games with definitions*

Ultralimits; close to section 2.4 of
Kleiner--Leeb, Rigidity of quasi-isometries...

*Tangent space*

*Halbeisen's example*

*Gradient flow*

*On area of boundary*