Plurifine Potential Theory

An overview of the recent developments in plurifine potential theory.Comment: 18 pages, Conference on SCV on the occasion of Jozef Siciak's 80'th birthda

Value distribution and potential theory

We describe some results of value distribution theory of holomorphic curves and quasiregular maps, which are obtained using potential theory. Among the results discussed are: extensions of Picard's theorems to quasiregular maps between Riemannian manifolds, a version of the Second Main Theorem of Nevanlinna for curves in projective space and non-linear divisors, description of extremal functions in Nevanlinna theory and results related to Cartan's 1928 conjecture on holomorphic curves in the unit disc omitting hyperplanes

Two symmetry problems in potential theory

We consider two eliiptic overdetermined boundary value problems. There are variants on J. Serrin's 1971 classical results and having the same conclusion that the domains should be forcibly Euclidean balls.Comment: 5 page

Gravity as BF theory plus potential

Spin foam models of quantum gravity are based on Plebanski's formulation of general relativity as a constrained BF theory. We give an alternative formulation of gravity as BF theory plus a certain potential term for the B-field. When the potential is taken to be infinitely steep one recovers general relativity. For a generic potential the theory still describes gravity in that it propagates just two graviton polarizations. The arising class of theories is of the type amenable to spin foam quantization methods, and, we argue, may allow one to come to terms with renormalization in the spin foam context.Comment: 7 pages, published in Proceedings of the Second Workshop on Quantum Gravity and Noncommutative Geometry (Lisbon, Portugal

An Overdetermined Problem in Potential Theory

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.Comment: updated version. 20 pages, 3 figure