SLE loop measures
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SLE loop measures Dapeng Zhan1 Received: 5 July 2018 / Revised: 7 March 2020 / Accepted: 11 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLEκ loop measures for κ ∈ (0, 8). First, we construct rooted SLEκ loop measures in the Riemann sphere C, which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its (1 + κ8 )-dimensional Minkowski content. Second, by integrating rooted SLEκ loop measures, we construct the unrooted SLEκ loop measure in C, which satisfies Möbius invariance and reversibility. Third, we use Brownian loop meaC to subdomains sures to extend the rooted and unrooted SLEκ loop measures from of C, which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted SLEκ loop measures to some Riemann surfaces. Finally, using a similar approach, we construct SLEκ bubble measures in simply/multiply connected domains rooted at a boundary point. The C confirms a conjecture by space-time homogeneity of rooted SLEκ loop measures in Greg Lawler on the existence of such measures. Mathematics Subject Classification 60D · 30C
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Symbols and notation . . . . . . . . . . . . . . . . . 2.2 SLE processes and their conformal Markov properties 2.3 Minkowski content measure . . . . . . . . . . . . . 2.4 Decomposition of chordal SLE . . . . . . . . . . . .
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Research partially supported by NSF Grant DMS-1056840 and Simons Foundation Grant #396973.
B 1
Dapeng Zhan [email protected] Michigan State University, East Lansing, USA
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D. Zhan 3 Whole-plane SLE under conformal distortion . . ˆ . . . . . . . . . . . . . 4 SLE loop measures in C 5 SLE loop measures in Riemann surfaces . . . . 6 SLE bubble measures . . . . . . . . . . . . . . A Chordal SLE in multiply connected domains . . B Image of an interior hull under a conformal map References . . . . . . . . . . . . . . . . . . . . . .
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