Brennan's conjecture and the integral-means spectrum

The explanation

Take a round disk of radius 1 and reshape it without tearing or folding. Such a map is called conformal when it keeps angles. Near each point it enlarges tiny shapes by one factor, called the stretch factor. A standard example is the Koebe map, f(z) = z/(1 − z)². Near the edge point 1 it stretches enormously. At z = 0.99 the factor is about 1,990,000. Near the opposite edge point −1 it squeezes instead. At z = −0.99 the factor is about 0.00127, a squeeze of about 788 times.

Squeezing is the hard part of this subject. One way to measure it is to walk round a circle of radius r inside the disk. At each point, take 1 divided by the squared stretch factor. Then average around the circle. For the Koebe map this average is about 225 at r = 0.9. It is about 3,092 at r = 0.99 and about 31,890 at r = 0.999. Multiply each by the distance to the edge, 1 − r. The results are about 22.5, 30.9 and 31.9, settling near a constant. So the average grows like 1 divided by the distance to the edge. Experts call the power in that rule, here 1, the growth exponent.

Brennan's conjecture runs such maps backwards. Start with any region in the plane that has no holes. Map it onto the disk with a conformal map φ (phi). Raise its stretch factor to a power s and add it up over the whole region. Is the total finite? The conjecture dates to James Brennan's work in 1978. It says yes for every s strictly between 4/3 and 4. The region may be unbounded, and its edge may be as rough as you like.

A change of variables turns this into a question about maps out of the disk. The power s becomes a new power t = 2 − s. The top end, s = 4, becomes t = −2. That is the squared squeeze from the example. The paper shows the Koebe map makes the total infinite at both ends, s = 4 and s = 4/3Source 1. So the range cannot be widened.

Progress came in pieces. Brennan's own work pushed the top of the range beyond 3. A 1999 thesis by Daniel Bertilsson reports the range from 4/3 to 3.421. Attention turned to a related number, the integral-means spectrum at −2. It is the largest growth exponent of the example's average, over all normalized maps of the disk. Normalized means the map keeps the center fixed and does not stretch it. The paper ties the value 1 at −2 to Brennan's conjectureSource 2. In 1994 Lennart Carleson and Nikolai Makarov found the spectrum for all sufficiently negative powers. Their result does not reach −2. Håkan Hedenmalm and Serguei Shimorin bounded the value at −2 by 1.218 in 2005. Alan Sola lowered it to 1.206 in a 2007 thesis. Recent papers proved the conjectured range for special families of maps.

Why was 1 so hard to reach? A classical estimate caps the squeeze at each single point. Squaring that cap gives growth exponent 2, not 1Source 3. The missing information is how often very strong squeezing can happen around one circleSource 4. The paper argues by contradiction. It supposes some maps grow faster than exponent 1. It turns that excess into a random way of choosing maps with exact scaling rulesSource 5. Those rules force a certain average to be positive. A second, geometric count forces the same average to be zero or belowSource 6. That count pairs each saddle point of a related height function with a distinct peak at least as highSource 7. The two signs clash, so the excess growth cannot exist.

The paper proves the full conjecture: the total is finite for every s between 4/3 and 4Source 8. It also proves that the spectrum at −2 is exactly 1Source 9. One constant works for every normalized map at once, at the cost of an arbitrarily small loss in the exponentSource 10. Back in the example, the Koebe average grew like 1 divided by the distance to the edge. No normalized map does worse, apart from that small loss. The paper also claims the spectrum equals |t| − 1 for every power t at or below −2Source 11.

A companion paper studies maps onto bounded regions at the power −1. Philipp Kraetzer's computer experiments on such maps led to a predicted formula. Its quadratic part appears in his 1996 paper and gives exactly 1/4 at −1. Earlier general bounds at −1 were 0.403, then 0.388. The companion proves a uniform bound with exponent strictly below 1/4Source 12. So the bounded spectrum at −1 is below 1/4, and Kraetzer's formula is false there. The proof shows that a gap exists but gives no number for its sizeSource 13. It does not settle the predicted value at the power +1Source 14.

How sure is this? A computer proof checker called Lean has checked formal versions of the main statements. They cover Brennan's range, the uniform bound at −2 and the value 1. They also cover the Koebe failures at the end points, and the companion's bound below 1/4. The Lean scope note does not mention the formula |t| − 1 for powers below −2. No specialist review has been published.

What it doesn't show. This does not give a number for the companion's gap below 1/4. It also leaves the predicted value at power +1 open.