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Critical measures in the complex plane are saddle points for the logarithmicenergy with external field. Their local and global structure was described byMartinez-Finkelshtein and Rakhmanov. In this paper we start the development ofa theory of critical measures on higher genus Riemann surfaces, where thelogarithmic energy is replaced by the energy with respect to a bipolar Green'skernel. We study a max-min problem for the bipolar Green's energy with externalfields Re V where dV is a meromorphic differential. Under reasonableassumptions the max-min problem has a solution and we show that thecorresponding equilibrium measure is a critical measure in the external field.In a special genus one situation we are able to show that the critical measureis supported on maximal trajectories of a meromorphic quadratic differential.  We are motivated by applications to random lozenge tilings of a hexagon withperiodic weightings. Correlations in these models are expressible in terms ofmatrix valued orthogonal polynomials. The matrix orthogonality is interpretedas (partial) scalar orthogonality on a Riemann surface. The theory of criticalmeasures will be useful for the asymptotic analysis of a correspondingRiemann-Hilbert problem as we outline in the paper.

Critical measures on higher genus Riemann surfaces