Brun theorem

2020 Mathematics Subject Classification: Primary: 53A04 Secondary: 53A35 [MSN][ZBL]

The problem and first results

The Painlevé problem is to find a characterization in geometric terms for the removable singularities of bounded analytic functions, or equivalently, for the null-sets of the analytic capacity. P. Painlevé studied this problem already in 1888 and proved a sufficient condition: if a compact plane set $K$ has length (that is, one-dimensional Hausdorff measure) zero, then it is removable for bounded analytic functions. The latter means that whenever $U$ is an open set in $\mathbb C$ containing $K$ and $f$ is bounded and analytic in $U\setminus K$, then $f$ has an analytic extension to $U$. In the other direction; if $K$ has Hausdorff dimension greater than 1 (in particular, if $K$ has interior points), then $K$ is not removable. A deep result due to A.P. Calder\'on from 1977 says that if $K$ is a subset of a rectifiable curve, then $K$ is removable if and only if it has length zero.

Tolsa's solution

In [T] X. Tolsa solved Painlevé's problem. The solution depends on the so-called Menger curvature $c(z_1,z_2,z_3)$ for triples of points in $\mathbb C$ and a formula of M.S. Melnikov relating it to the Cauchy kernel $1/z$. By definition the Menger curvature is the reciprocal of the radius of the circle passing through the points $z_1,z_2,z_3$; it is equal to zero if and only if the three points lie on one line. Tolsa's solution is:

A compact set $K\subset \mathbb C$ is not removable for bounded analytic functions if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that $\mu(D)\leq diam(D)$ for all discs $D$ in $\mathbb C$ and $\int\int\int c(z_1,z_2,z_3)^2d\mu z_1d\mu z_2d\mu z_3<\infty$.

Good general reference is [P].

References

[P]	H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002.
[T]	X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica, 190 (2003), 105-149.