# MATH 86100: Metric Geometry

Thursdays 11:45-1:45 pm

Room 5417

Professor: Christina Sormani

Topics: Lipschitz Maps, Hausdorff Measure and Dimension, Length Spaces, Hyperbolic Space, Alexandrov Spaces, Fundamental Groups of these Spaces, Gromov-Hausdorff Convergence, Gromov's Compactness Theorem, Quasi Isometries, Gromov Hyperbolic Spaces, Gromov-Hausdorff Limits, Tangent Cones at Infinity, and Metric Measure Spaces. Applications to Riemannian Geometry, the work of Perelman and Complex Analysis will be discussed.

Text: Burago-Burago-Ivanov's "A Course in Metric Geometry" AMS Graduate Studies in Mathematics Volume 33. See this link for typos. It's easiest to order this book from the AMS.

Prerequisites: Graduate Topology and Real Analysis. Differential Geometry is a corequisite.

Meetings:

• Sept 1: An Introduction
Covered 1.1-1.6 of BBI, defined quasi-isometries and epsilon almost isometries and epsilon nets. Be sure to read 1.3 on your own.
• Sept 8: Compact Spaces, Hausdorff Measures and Dimensions
BBI 1.6-1.7 and Morgan's Geometric Measure Chap 1
Homework assigned from BBI Ch 1 due on Sept 15: 1.4.9, 1.5.2, 1.5.8, 1.6.10, 1.6.13,
• Sept 15: Length Structures, Strictly Intrinsic Metrics BBI Chapter 2.
Read 2.1-6, Do exercises 2.1.5, 2.1.8, 2.1.9, 2.1.2 and rewrite the proof of Lemma 2.6.1 with appropriate pictures.
• Sept 22: Metric Measure Spaces and Ricci Curvature
to prepare students for Max von Renesse's Talks on Tuesday Sept 27.
Handwritten lecture notes for today in pdf contains quite a few exercises.

A nicely written proof of the Volume Comparison Theorem by Zhu is here
• Sept 29: Constructions of Spaces: Gluing and Covering Spaces
I will cover 3.1, 3.3, 3.4 in class, (3.2 will be next week)
The definition of delta covers can be found in an article by Guofang Wei and I in AMS Transactions and as a preprint in the arxiv.
The Covering Spectrum can be found in our JDG paper also available as a preprint in the arxiv.

Read 3.1, 3.3 and 3.4/ Do 3.1.13, 3.1.14, 3.1.19, 3.3.9, 3.4.6, 3.4.7, 3.4.8, 3.4.9,
• Oct 6: Quasi Isometries and Graphs
Christine Abreu Suzuki will present her thesis on quasi isometries during the first hour. ([BBI] 8.3)
Jozef Dodziuk will present Cayley graphs, word metrics and combinatorial Laplacians the second hour. ([BBI] 3.2)
Read 3.2, Do 3.2.12, 3.2.19, 3.2.21, 3.2.24 Read 8.3, Do 8.3.7, 8.3.8, 8.3.13, 8.3.16, 8.3.17, 8.3.22
• Oct 13: Yom Kippur
• Oct 20: Cones, Warped Products, and Spaces of Bounded Curvature,
BBI 3.6 and 4.1 Read 3.6: Do 3.6.3 note the tori are taxicab here, 3.6.23, show the standard sphere is the circle warped by sin(t) over the interval [0,pi], show the hyperbolic plane is the circle warped by sinh(t) over the interval [0, infty), 3.6.24, show that any radially symmetric surface
x 2 + y2 = r(z) 2 with r(z)>0
is a warped product of a line with a circle with a warping function
f(t) such that f(t(z))=r(z) when t(z) is arclength of (z, r(z)) from 0 to z.
Do 3.6.24, 3.6.28, 3.6.29, Prove Prop 3.6.30, Do 3.6.36, 3.6.40,
• Oct 27: Chap 4 Alexandrov Spaces
Two important recent construction articles by Alexander and Bishop are posted here and here for future reading. A nice survey of the Alexandrov Spaces by Otsu can be found here.
Read 4.1-3, Do 4.1.7, 4.1.11. 4.1.14, 4.2.8, 4.2.10, 4.3.8
• Nov 3: Class Cancelled (extra class added Tue Nov 22 2pm)
• Nov 10: Chapter 7 Gromov's Compactness Theorem
We will review various equivalent definitions of Gromov-Hausdorff Convergence and then prove Gromov's Precompactness Theorem (7.4.15) mentioned on Sept 22 and prove that Gromov-Hausdorff limits of length spaces are length spaces (7.5.1). Together, these form Gromov's Compactness Theorem. See also Gromov's "Metric Structures..." Chapter 5.
Homework: Read Chapter 7 except the last subsection 7.5.2 on the topology of Gromov-Hausdorff limits which has many errors, do 7.3.2, 7.3.4, 7.3.5, 7.3.13, 7.3.14, 7.3.15, 7.3.19, 7.3.23, 7.3.24, 7.3.26, 7.3.31,
• Nov 17: 7.5.1: Gromov-Hausdorff limits: when they are compact, when they are length spaces, and how every length space can be achieved as a limit of graphs
Homework: 7.4.5, 7.4.6, 7.4.14, 7.4.17
Section 7.5.9 on the topology of limit spaces has serious errors, see instead:
"Hausdorff Convergence and Universal Covers" by Sormani and Wei in Transactions of the American Mathematical Society 353 (2001), no. 9, 3585--3602. One can prove BBI Exer 7.5.9 using these techniques, but BBI Exer 7.5.11 is false. There is an example of compact simply connected length spaces Xi which converge to a compact length space that is not simply connected in this Sormani-Wei paper and also elsewhere. The correction to Exer 7.5.11 given on Y. Burago's website is to assume the limit space \$X\$ is locally simply connected in which case students who read this paper should be able to complete the exercise.
• Nov 22: (Tuesday 2pm Makeup lesson in Room 8405) A discussion of currents (see Frank Morgan's Geometric Measure Theory)
At 4pm Stefan Wenger (CIMS) will speak on Alexandrov Spaces for the Differential Geometry Seminar. Tea will be served at 3:30pm in the math lounge.
• Nov 24: Thanksgiving
• Dec 1: Tangent Cones at Infinity and Structure
8.1-2, a discussion of work by Cheeger-Colding, Sormani-Wei, Menguy, Otsu
Homework: 7.3.6,
• Dec 6 (Tuesday): Talk by Kenneth Clarkson (Bell Labs) 2:30 pm Differential Geometry Seminar on metric spaces applied to real world problems.
• Dec 8: Convergence of Geodesics and the Covering Spectrum and a quick review of the Hyperbolic Space
See 5.3, particularly pages 174-5, for a review of the Hyperbolic Plane, but beware some serious typos. You may wish to consult other sources. The most serios typo is on page 175 line 11 should say:
d(a,p)+d(b,p) < d(a,b) +4 or d(a,p) + d(c,p) < d(a,c) +4
Notice both the points are permuted as well as 4 instead of a 2. This is easy to prove using the geodesic [bc] is in the tubular neighborhood T2([ab] union [ac]).
See "The Covering Spectrum of a Compact Length Space" by Sormani and Wei, Journal of Differential Geometry 66 (2004) 647-689.
• Dec 15: Gromov Hyperbolic Spaces Cancelled Makeup in the Spring
8.4 in [BBI] is vague and has many typos. On page 286 he uses the term delta almost degenerate without being careful to label that it is relative to p and then lines 9 and 10 need opposite inequalities. I am going through this carefully and will post more corrections as we proceed. A nice alternate reference is Kapovich's Lectures on Geometric Group Theory. Gromov Hyperbolic Spaces" by Vaisala has a nice exposition when one assumes the space is only intrinsic.
• Final Assignment (due Thursday Dec 22 at 11:45 am): The final assignment is a collection of 20 problems selected from the homework completed, submitted and resubmitted during the semester as listed below. Please gather it into a single packet (possibly including extra problems) and add a cover sheet with the list of which 10 correctly completed problems should be used for the grade:
• Four out of the five problems assigned on Sept 8,
• Four out of the five problems assigned on Sept 15,
• Four of the eight problems assigned on Sept 29
• One problem from 3.2 and two from 8.3 assigned on Oct 6
• Two of the problems from Oct 20
• One problem from Oct 27
• Two of the problems from Nov 10 - 17.
A student may replace any problem with 2 different problems selected by the student. So if you choose to skip the two problems from 8.3, you may do four extra from elsewhere. To keep this under control, lets max this out at five problems replaced by 10.

Be careful to explain you've done this on the cover sheet.

The problems will be given grades of 0-5 based on correctness of the proofs with the usual grades of A - F given as follows:
A 90-100, B 80-90, C 70-80, D 60-70, F 0-60