The Loewner differential equation and slit mappings

By Donald E. Marshall, Steffen Rohde. 2005

130 citations recorded by the source corpus.

In his study of extremal problems for univalent functions, K. Lowner [11] (who later changed his name into C. Loewner) introduced the differential equation named after him. It was a key ingredient in the proof of the Bieberbach conjecture by de Branges [2]. It was used by L. Carleson and N. Makarov in their investigation of a process similar to DLA [3]. Recently, O. Schramm [20] found a description of the scaling limits of some stochastic processes in terms of the Loewner equation (assuming the validity of some conjectures such as existence of the limits). This led him to the definition of a new stochastic process, the “Stochastic Loewner Evolution” (SLE). The SLE has been further explored in the work of Lawler, Schramm and Werner (see [14] and the references therein) and led to a proof of Mandelbrot’s conjecture about the Hausdorff dimension of the Brownian frontier. The SLE has also played a crucial role in S. Smirnov’s [21] recent work on percolation. Somewhat surprisingly, the geometry of the solutions to the Loewner equation is not very well understood. This paper addresses the regularity of solutions in the deterministic setting. Let Ωt, t0 ≤ t ≤ t1, be a continuously increasing sequence of simply connected planar domains, and let z0 ∈ ⋂ t Ωt. Then there are conformal maps ft : D → Ωt of the unit disc with ft(0) = z0 and f ′ t(0) > 0. The continuity of the domain sequence can be expressed by saying that the map t → ft is continuous in the topology of locally uniform convergence. Since f ′ t(0) is increasing in t, we may assume (by reparametrizing Ωt if necessary) that f ′ t(0) = e t for t0 ≤ t ≤ t1. The family ft, t0 ≤ t ≤ t1, is called a (normalized) Loewner chain. The Loewner differential equation

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5404cf4b633bfaa95ebabea0740faac2feb430af

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In Applied Mathematics

1. Manuel D. Contreras
2. S. Díaz-Madrigal
3. B. Filippo
4. Yilin Wang
5. Joan Lind
6. Filippo Bracci

In Mathematical Physics

1. Manuel D. Contreras
2. S. Díaz-Madrigal
3. Yilin Wang
4. B. Filippo
5. Joan Lind
6. P. Gumenyuk
7. Filippo Bracci

Citations in the corpus

• A Sharp Condition for the Loewner Equation to Generate Slits (2003, methodology): “Rohde’s recent work [11] concerning when the Loewner equation can generate slits.”
• Evolution families and the Loewner equation I: the unit disc (2008, background): “Also Bauer [2], Marshall and Rohde [18] and Prokhorov and Vasiliev [23] studied a similar chordal Loewner equation.”
• Loewner chains in the unit disk (2009, background): “Some recent developments concerning the relationship between properties of the driving term and the geometry of solutions to (1.4) and (1.5) can be found in [19, 26 , 30].”
• The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “For instance, Lind, Marshall and Rohde (see [17]) exhibit a driving function with λ(t) ∼ 4 √ 1− t when t→ 1−, that generates a Loewner chain with infinite spiral at time 1−.”