Question

Hey Folks!

Thanks to Jasun for posting all the notes. They're great. I hope you all enjoyed the workshop too - I was really jealous.

So here's the question. I haven't done much research on this, so I don't know if it's impossible or totally trivial.

A theorem of Ahlfors and Beurling* states that every quasisymmetric map of the circle extends to a quasisymmetric map of the disk. That is, every quasicircle is the boundary of a quasidisk. Now, every bi-Lipschitz map is quasistymmetric, so every bi-Lipschitz image of the circle is the boundary of a quasidisk. But, does every bi-Lipschitz map of the circle extend to a bi-Lipschitz map of the disk?

The obvious thing to try would be to see if the Ahlfors-Beurling** extension for quasisymmetric maps produces a bi-Lipschitz map from bi-Lipschitz boundary data. However, I don't really remember how the Ahlfors-Beurling*** extension works, and since I'm on the road, I can't really look it up (a technical version is in Lehto's book, but Mario explained to me once the "right" way of thinking about it - damn my crappy memory!) Anyway, any insights would be appreciated.

Another way to think about this would be to use some Lipschitz extension theorems. However, the McShane extension wouldn't work because there is no way in that construction to ensure that the extension of a bi-lip homeo is a bi-lip homeo. (one only gets Lipschitz, not bi-Lipschitz, which is a result of the fact that one must apply the extension to the coordinate functions individualy). Maybe Kirzbaum is better? (again, I'm on the road so I can't look up the proof of kirzbaums theorem).

Ok, thanks!

*Thanks to Anonymus for correcting this reference.
**Ditto
***Ditto Ditto

last licks: a belated update.

the last five lectures (32-36) are available at the usual place. the very last lectures have the flavor of a seminar talk, and for more depth or details, refer to the ICM preprints of M. Bonk and B. Kleiner.

sorry for the delay; after classes ended, i was hardly anywhere less than 10 meters from a coffee stand. q:

belated update.

seven (7) new lectures available in PDF at the usual place. sorry for the delay.

to clarify matters, every monday or so is a double session of student talks for grade assessment. i've chosen not to take notes during these talks, but happily every speaker thus far has made LaTeX PDF handouts for the audience. if you would like some, then go bug these people:

a review of basic riemannian geometry (jason miller)

in which we examine the riemann curvature tensor through connections, and compute the sectional curvature of hyperbolic n-space to be constant (-1).

on liouville's theorem (aaron magid)

this is a proof of a rigidity theorem: given a domain in euclidean n-space (n ≥ 3) a thrice-differentiable conformal transformation is the restriction of a möbius transformation. the proof rests on differentiating to obtain an overdetermined system of pde, where our computations are motivated by appropriate curvature quantities from riemannian geometry.

the hyperboloid model for hyperbolic space (karl weintraub)

this is a construction of an equivalent model for hyperbolic n-space, using the lorentz metric on a light cone, defining an appropriate projection, and demonstrating that it is isometric to the disc model.

on the measurable riemann mapping theorem (marie snipes)

the outline: a quick introduction to sobolev spaces and quasiconformal mappings, then formulating the beltrami equation, and through means of clever singular integral operators (the cauchy and hilbert transforms), one proves that given two simply connected domains in the plane and a beltrami differential, there is a quasiconformal homeomorphism of the prescribed domains and with the prescribed dilatation.

FTP question

Why is that ever since we migrated from math.lsa.umich.edu to itd.umich.edu, I have been unable to use sftp to get to my file space?

Before, from any terminal I could run sftp name@login.math.lsa.umich.edu and get to my space, but a similar command sftp name@login.itd.umich.edu yields an error "connection reset by peer". I get the same error using the JYU network as I do from my laptop.

Any ideas how I can fix this?

cool theorem

Consider the following game:

Let epsilon > 0. Let U be a domain in the plane, and define "the ball" to be a point x in U. There are two players, A and B. A turn consists of

1) Player A picks a unit vector v.
2) Player B sets b= +/- 1.
3) The ball is moved from x to x+b*sqrt(2)*epsilon*v.

The goal of A is to move the ball outside of U. The goal of B is prevent A from doing so.

It is not to hard to show that A can always win. In fact, if u_ep(x) = (epsilon)^2 k where k is the minimum number of turns in the optimal winning strategy for A and x is the starting point of the ball, then u_ep is bounded independently of x depending only the diameter of U. What is really interesting is that the function u defined by

u(x)=lim_(epsilon ->0)u_ep(x)

satisfies the pde

|grad(u)|div((grad u)/|grad u|) = -1 on U
u=0 on bdry U.

This is the same pde that governs the mean curvature flow. Creepy.

after long last ..

there are five (5) new lectures available here. i apologize for the delay; after spring break i forgot to stop by shapiro library and scan the lot of them.

one lecture has no theorems in bold ink; a classmate's pen ran out of ink, and he borrowed mine.

Question

Hi Folks!

Sorry I've been out of touch. You know, I've been trying to integrate myself (was area man, now volume man, plus constant). Anyway, I haven't had enough coffee, and I seem to proving contradictions in mathematics. Can anyone verify or debunk the following seemingly obvious statement? I have hopelessly confused myself.

Let C be the 1/3 Cantor set. Let s=log2/log3. Then Hausdorff s-measure(C)=1=Hausdorff s-content(C).

Right? Philosophy: For the Cantor set, the obvious cover is the most efficient. Of coursing proving this is a pain - see notes from Juha's course last semester.

Am I crazy?