Equilibrium Problems of Potential Theory in the Complex Plane Andrei Mart´ ınez Finkelshtein ⋆ Departamento de Estad´ ıstica y Matem´atica Aplicada, Universidad de Almer´ ıa, 04120, Almer´ ıa, Spain and Instituto Carlos I de Fisica Te´orica y Computacional, Universidad de Granada, Spain e-mail:andrei@ual.es Summary. This is a short introduction to the theory of the logarithmic potential in the complex plane. The central ideas are the concepts of energy and equilibrium. We prove some classical results characterizing the equilibrium distribution and dis- cuss the extension of these notions to more general settings when an external field or constraints on the distribution are present. The tools provided by potential the- ory have a profound impact on different branches of analysis. We illustrate these applications with two examples from approximation theory and complex dynamics. 1 Background ............................................... 80 1.1 Introduction ........................................... 80 1.2 Background or What You Should Bring to Class ............ 80 2 Logarithmic Potentials: Definition and Properties ......... 82 2.1 Superharmonic Functions ................................ 82 2.2 Definition of the Logarithmic Potential .................... 84 2.3 Some Principles for Potentials ............................ 85 2.4 Recovering a Measure from its Potential ................... 86 3 Energy and Equilibrium ................................... 88 3.1 Logarithmic Energy ..................................... 88 3.2 Extremal Problem, Equilibrium Measure and Capacity ...... 90 3.3 Link with Conformal Mapping and Green Function ......... 96 3.4 Equilibrium in an External Field ......................... 100 3.5 Other Equilibrium Problems. Equilibrium with Constraints . . 106 ⋆ Supported, in part, by a research grant from the Ministry of Sciences and Technol- ogy (MCYT) of Spain (MTM2005-08648-C02-01), by Junta de Andaluc´ ıa, Grupo de Investigaci´on FQM229, by INTAS Research Network on Constructive Complex Approximation (INTAS 03-51-6637), and by NATO Collaborative Linkage Grant PST.CLG.979738.