arXiv subjects

Andrea del Monaco

Publications and source records attributed to Andrea del Monaco.

3 recordsLinked to original sources

arXiv2016

Tightness results for infinite-slit limits of the chordal Loewner equation

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting $N$ points on $\mathbb{R}$ to infinity within the upper half-plane. For every $N\in\mathbb{N}$, this equation provides a measure valued process $t\mapsto \{α_{N,t}\},$ and we are interested in the limit behaviour as $N\to\infty.$ We prove tightness of the sequence $\{α_{N,t}\}_{N\in\mathbb{N}}$ under certain assumptions and address some further problems.

math.CV↗

arXiv2015

Multiple SLE and the complex Burgers equation

In this paper we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of $n$ slits that connect $n$ points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.

math.CV↗

arXiv2013

Chordal Loewner Equation

The aim of this survey paper is to present a complete direct proof of the well celebrated cornerstone result in Loewner Theory, originally due to Kufarev et al [Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 200 (1968) 142-164. MR0257336 (41 #1987)], stating that the family of the hydrodynamically normalized conformal self-maps of the upper-half plane onto the complement of a gradually erased slit satisfies, under a suitable parametrization, the chordal Loewner differential equation. The proof is based solely on basic theorems of Geometric Function Theory combined with some elementary topological facts and does not require any advanced technique.

math.CV↗