Laplacian Growth and Whitham Equations of Soliton Theory

The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian growth is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde

Growth of fat slits and dispersionless KP hierarchy

A "fat slit" is a compact domain in the upper half plane bounded by a curve with endpoints on the real axis and a segment of the real axis between them. We consider conformal maps of the upper half plane to the exterior of a fat slit parameterized by harmonic moments of the latter and show that they obey an infinite set of Lax equations for the dispersionless KP hierarchy. Deformation of a fat slit under changing a particular harmonic moment can be treated as a growth process similar to the Laplacian growth of domains in the whole plane. This construction extends the well known link between solutions to the dispersionless KP hierarchy and conformal maps of slit domains in the upper half plane and provides a new, large family of solutions.Comment: 26 pages, 6 figures, typos correcte

Topology of quadrature domains

We address the problem of topology of quadrature domains, namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity.Comment: 37 pages, 11 figures in J. Amer. Math. Soc., Published electronically: May 11, 201

Geometric Methods of Complex Analysis (hybrid meeting)

The purpose of this workshop was to discuss recent results in Several Complex Variables, Complex Geometry and Complex Dynamical Systems with a special focus on the exchange of ideas and methods among these areas. The main topics of the workshop included Holomorphic Dynamics and Nevanlinna's Theory; $L^2$-methods and Cohomologies; Plurisubharmonic Functions and Pluripotential Theory; Geometric Questions of Complex Analysis

From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this result to the case of a nef and big cohomology class is also obtained. As a consequence new PDE proofs are obtained for the regularity results for envelopes in [14] (which, however, are weaker than the results in [14] in the case of a non-nef big class). Applications to the regularization problem for quasi-psh functions and geodesic rays in the closure of the space of Kahler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of the regularizations as transcendental Bergman metrics.Comment: 28 pages. Version 2: 29 pages. Improved exposition, references updated. Version 3: 31 pages. A direct proof of the bound on the Monge-Amp\`ere mass of the envelope for a general big class has been included and Theorem 2.2 has been generalized to measures satisfying a Bernstein-Markov propert