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Logarithmic Potential Theory on Riemann Surfaces


Skinner, Brian Paul (2015) Logarithmic Potential Theory on Riemann Surfaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9Q52MK8.


We develop a logarithmic potential theory on Riemann surfaces which generalizes logarithmic potential theory on the complex plane. We show the existence of an equilibrium measure and examine its structure. This leads to a formula for the structure of the equilibrium measure which is new even in the plane. We then use our results to study quadrature domains, Laplacian growth, and Coulomb gas ensembles on Riemann surfaces. We prove that the complement of the support of the equilibrium measure satisfies a quadrature identity. Furthermore, our setup allows us to naturally realize weak solutions of Laplacian growth (for a general time-dependent source) as an evolution of the support of equilibrium measures. When applied to the Riemann sphere this approach unifies the known methods for generating interior and exterior Laplacian growth. We later narrow our focus to a special class of quadrature domains which we call Algebraic Quadrature Domains. We show that many of the properties of quadrature domains generalize to this setting. In particular, the boundary of an Algebraic Quadrature Domain is the inverse image of a planar algebraic curve under a meromorphic function. This makes the study of the topology of Algebraic Quadrature Domains an interesting problem. We briefly investigate this problem and then narrow our focus to the study of the topology of classical quadrature domains. We extend the results of Lee and Makarov and prove (for n ≥ 3) c ≤ 5n-5, where c and n denote the connectivity and degree of a (classical) quadrature domain. At the same time we obtain a new upper bound on the number of isolated points of the algebraic curve corresponding to the boundary and thus a new upper bound on the number of special points. In the final chapter we study Coulomb gas ensembles on Riemann surfaces.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Complex Analysis; Mathematical Physics; Logarithmic Potential Theory; Laplacian Growth; Coulomb Gas Ensembles
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Makarov, Nikolai G.
Thesis Committee:
  • Makarov, Nikolai G. (chair)
  • Kechris, Alexander S.
  • Marcolli, Matilde
  • Alberts, Tom
Defense Date:25 June 2014
Record Number:CaltechTHESIS:05292015-072640484
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8915
Deposited By: Brian Skinner
Deposited On:02 Jun 2015 15:25
Last Modified:04 Oct 2019 00:08

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