×

Convex shapes and harmonic caps. (English) Zbl 1382.53021

Let \(P\) be a planar shape, that is, \(P\) is a compact, connected subset of \(\mathbb C\) that contains at least two points and has connected complement. Let \(\mu\) be a probability measure on \(\partial P\).
The authors study the existence of a conformal metric \(\rho(P,\mu)=\rho(z)|dz|\) on the Riemann sphere \(\hat{\mathbb C}\) so that \(P\) embeds locally-isometrically into (\(\hat{\mathbb C},\rho\)) and the curvature distribution \(\omega_{\rho}=-\Delta\log\rho(z)\) is equal to the push-forward of \(4\pi\mu\) under the embedding. The complement \(\hat{P_{\mu}}\) of \(P\) in (\(\hat{\mathbb C},\rho\)) is called the cap of the pair \((P,\mu)\). When \(\mu\) is the harmonic measure of \(\hat{\mathbb C}\setminus P\) relative to \(\infty\), \(\hat{P_{\mu}}\) is called the harmonic cap.
In Theorem 1.1 the authors prove that a Euclidean development of a harmonic cap \(\hat{P_{\mu}}\) is given by a locally univalent function \(g:\mathbb D\to\mathbb C\) defined by \[ g(z)=\int_{0}^{z}\Phi'(1/w)dw, \] where \(\mathbb D\) is the open unit disc and \(\Phi:\hat{\mathbb C}\setminus\overline{\mathbb D}\to\hat{\mathbb C}\setminus P\) is a conformal isomorphism with \(\Phi(\infty)=\infty\). In Theorem 1.2 they prove that, if \(P\) is bounded by a piecewise differentiable Jordan curve and \(\mu\) is a measure on \(\partial P\) such that the cap \(\hat{P_{\mu}}\) exists, then the boundary identification between \(\partial P\) and \(\partial\hat{P_{\mu}}\) that produces (\(\hat{\mathbb C},\rho\)) is given by \(s(t)\sim \hat{s}(t)\), where \[ s(t)=\int_{0}^{t}e^{i\alpha(x)}dx \] is a counterclockwise, unit-speed parametrization of \(\partial P\) and \[ \hat{s}(t)=\int_{0}^{t}e^{i(\alpha(x)-4\pi\mu(s(0,t]))}dx. \] Moreover, the authors characterize the cases where the metric \(\rho(P,\mu)\) exists for shapes \(P\) bounded by Jordan curves and arbitrary probability measures \(\mu\) on \(\partial P\). Finally, they provide examples of harmonic caps coming from connected filled Julia sets of polynomials with illustrating figures.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
52A10 Convex sets in \(2\) dimensions (including convex curves)
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York (1973) · Zbl 0537.58035
[2] Aleksandrov, A.D.: Vnutrennyaya Geometriya Vypuklyh Poverhnosteĭ. OGIZ, Moscow (1948)
[3] Alexandroff, A.: Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S. 11(53), 15-65 (1942) · Zbl 0061.37603
[4] Alexandrov, A.D.: Convex polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005) (translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov) · Zbl 1067.52011
[5] Bartholdi, L.: Personal communication (2015)
[6] Bobenko, A.I., Izmestiev, I.: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447-505 (2008) · Zbl 1154.52005 · doi:10.5802/aif.2358
[7] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6(103-144), 1965 (1965) · Zbl 0127.03401
[8] Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30(2), 205-239 (2003) [U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000)] · Zbl 1046.52016
[9] Demaine, E.D., O’Rourke, J.: A survey of folding and unfolding in computational geometry. In: Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ., vol. 52, pp. 167-211. Cambridge Univ. Press, Cambridge (2005) · Zbl 1094.70003
[10] Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms. Cambridge University Press, Cambridge (2007) (Linkages, origami, polyhedra) · Zbl 1135.52009
[11] DeMarco, L.: Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326(1), 43-73 (2003) · Zbl 1032.37029 · doi:10.1007/s00208-002-0404-7
[12] Duren, P.L.: Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer, New York (1983) · Zbl 0514.30001
[13] Fornæss, J.E., Sibony, N.: Complex dynamics in higher dimensions. In: Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, pp. 131-186. Kluwer Acad. Publ., Dordrecht (1994) (Notes partially written by Estela A. Gavosto) · Zbl 0811.32019
[14] Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45-62 (1983) · Zbl 0568.58027 · doi:10.1007/BF02584744
[15] Hubbard, J., Papadopol, P.: Superattractive fixed points in \[{ c}^n\] cn. Indiana Univ. Math. J. 43, 321-365 (1994) · Zbl 0858.32023 · doi:10.1512/iumj.1994.43.43014
[16] Hubbard, J.H.: Teichmüller theory and applications to geometry, topology, and dynamics, vol. 1. Matrix Editions, Ithaca (2006). Teichmüller theory, with contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, with forewords by William Thurston and Clifford Earle · Zbl 1102.30001
[17] Ju, M.: Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3(3), 351-385 (1983) · Zbl 0537.58035
[18] Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. American Mathematical Society, Providence (1973) (translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 35) · Zbl 0311.53067
[19] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992) · Zbl 0762.30001
[20] Rešetnjak, Ju.G.: Isothermal coordinates on manifolds of bounded curvature. I, II. Sibirsk. Mat. Ž. 1, 88-116, 248-276 (1960)
[21] Reshetnyak, Yu.G.: Two-dimensional manifolds of bounded curvature. In: Geometry, IV, Encyclopedia Math. Sci., vol. 70 , pp. 3-163, 245-250. Springer, Berlin (1993) · Zbl 0781.53050
[22] Series, C.: Thurston’s bending measure conjecture for once punctured torus groups. In: Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser., vol. 329, pp. 75-89. Cambridge Univ. Press, Cambridge (2006) · Zbl 1104.30029
[23] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 2nd edn. Publish or Perish Inc, Wilmington (1979) · Zbl 0439.53002
[24] Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, Geom. Topol. Monogr., vol. 1, pp. 511-549. Geom. Topol. Publ., Coventry (1998) · Zbl 0931.57010
[25] The Sage Developers. Sage Mathematics Software (Version 6.10.beta7) (2016). http://www.sagemath.org
[26] Van Andel, E., Bradshaw, R.: Riemann mapping [sage mathematics software package] (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.