Critical Measures on Higher Genus Riemann Surfaces

Abstract

Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martínez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green’s kernel. We study a max-min problem for the bipolar Green’s energy with external fields $\text { Re }V$ where dV is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann–Hilbert problem as we outline in the paper.

A Existence and Symmetry of the Bipolar Green’s Function

Let X be a Riemann surface and let $p_\infty$ be a distinguished point at infinity. Our goal is to prove Proposition 2.1 and in particular part (d) that gives us the symmetry $G(p,q) = G(q,p)$ of the bipolar Green’s function. For this proof we rely on Riemannian geometry where the existence of a symmetric Green’s function for the Laplacian is known.

1.1 A.1 Riemann surfaces as Riemannian manifolds

We first recall some relevant notions from Riemannian geometry. The main reference in this section is [38]. As shown in Lemma 2.3.3 in [38], X admits a conformal Riemannian metric $\rho$, which is given in the local coordinate (U, z) by

$$\begin{aligned} \rho _U(z)^2 dz d\overline{z}, \quad \rho _U(z) > 0, \end{aligned}$$

where $\rho _U$ is smooth and which transforms correctly under a holomorphic change of local coordinates. This turns X into a Riemannian manifold. The Riemannian metric allows for the definition of the length of a curve $\gamma \subset U$ and area of a measurable set $B \subset U$, namely

$$\begin{aligned} \ell (\gamma ):= \int _\gamma \rho _U(z) |dz|, \quad {\text {area}}(B):= \frac{i}{2} \int _B \rho _U(z)^2 dz d\overline{z}; \end{aligned}$$

see also [38, p. 21]. The length of general curves is computed by splitting the curve into a finite number of pieces, each of which is contained in a single coordinate chart; a similar method works for the area. Note that ${\text {area}}(X)$ is finite as X is compact.

The Riemannian metric $\rho$ globally defines an area form (i.e., a nowhere-vanishing 2 form) dA (the dependence on $\rho$ is suppressed in the notation) in the local coordinate (U, z) by

$$\begin{aligned} \frac{i}{2} \rho _U(z)^2 dz d\overline{z}. \end{aligned}$$

The area form dA will be used to integrate functions.

The distance $d: X \times X \rightarrow [0,+\infty )$ between two points p and q can then be defined as

$$\begin{aligned} d(p,q):= \inf \left\{ \ell (\gamma ) \mid \gamma : [0,1] \rightarrow X \text { is a curve with }\gamma (0) = p \text { and }\gamma (1) = q \right\} .\nonumber \\ \end{aligned}$$

(A.1)

The metric topology on X defined by d coincides with the original topology on X. This is because in a coordinate chart (U, z), the induced distance d(z(p), z(q)) is equivalent to the Euclidean distance in z(U) in the sense that for a fixed compact set $K \subset U$,

$$\begin{aligned} c |z(p) - z(q)| \le d(z(p),z(q)) \le C |z(p) - z(q)|, \quad p,q \in K, \end{aligned}$$

(A.2)

where $c > 0$ and $C > 0$ are the minimum and maximum of $\rho _U$ on K respectively. See also [1, Theorem 1.18].

Lemma A.1

Let $K \subset X$ be compact and let $\{U_j\}_{j=1}^n$ be a finite open cover of K with coordinate charts $(U_j, z_j)$. Then there exists an $\eta > 0$ and a $C > 0$ such that for every subset $K'$ of K with diameter at most $\eta$, $K' \subset U_j$ for some j and

$$\begin{aligned} {{\,\textrm{diam}\,}}(K') \le C {{\,\textrm{diam}\,}}(z_j(K')). \end{aligned}$$

Note that we have the diameter with respect to d (see (A.1)) on the left and the standard Euclidean diameter on the right.

Proof of Lemma A.1

Let $K \subset X$ be compact and let $\{U_j\}_{j=1}^n$ be a finite open cover of K with coordinate charts $(U_j, z_j)$ be given. By the properties of the metric topology, we can find open sets $\tilde{U}_j$ that are compactly contained in $U_j$ (that is, the closure of $\tilde{U}_j$ is compact and contained in $U_j$) such that $K \subset \cup _{j=1}^n \tilde{U}_j$. Hence by (A.2), for every $j = 1, \ldots , n$, there are numbers $C_j > 0$ such that

$$\begin{aligned} d(p,q) \le C_j |z_j(p) - z_j(q)|, \quad p,q \in \tilde{U}_j. \end{aligned}$$

Let $\eta > 0$ be a Lebesgue number for the open cover $\{\tilde{U}_j\}_{j=1}^n$ of K (so every subset of K with diameter at most $\eta$ is contained in some $\tilde{U}_j$) and take $C = \max _j C_j > 0$. Suppose that $K' \subset K$ has diameter ${{\,\textrm{diam}\,}}(K') \le \eta$. Then $K'$ lies in some $\tilde{U}_j$. Moreover,

$$\begin{aligned} {{\,\textrm{diam}\,}}(K')&= \sup _{p,q \in K'} d(p,q) \le C_j \sup _{p,q \in K'} |z_j(p) - z_j(q)| \\&\le C \sup _{p,q \in K'} |z_j(p) - z_j(q)| = C {{\,\textrm{diam}\,}}(z_j(K')), \end{aligned}$$

which concludes the proof. $\quad \square$

The Riemannian metric $\rho$ moreover defines the Laplace-Beltrami operator $\Delta$ on functions $f \in C^2(X)$ by locally setting

$$\begin{aligned} \Delta f = \frac{4}{\rho _U(z)^2} \frac{\partial }{\partial z} \frac{\partial }{\partial \overline{z}} f, \end{aligned}$$

see [38, Definition 2.3.3] and also [20, Section 1.1]. Note that $\Delta f$ is a function on X. Moreover,

$$\begin{aligned} \Delta f = - \star d \star df \end{aligned}$$

where $\star$ denotes the Hodge star operator on k-forms (see [38, Section 5.2]). The 2-form $d \star df$ is independent of $\rho$ (and is sometimes taken as a definition for the Laplacian, see e.g. [28]) and hence $\Delta$ only depends on $\rho$ through the application of the Hodge star operator on $d \star d f$.

1.2 A.2 Proof of Proposition 2.1

Because X can be turned into a compact Riemannian manifold, it carries a Green’s function of the Laplacian [1, Theorem 4.13] (see also [4, Theorem 2.1]). This is a real-valued function $\tilde{G}$ defined on $X \times X$ minus the diagonal that is smooth, symmetric

$$\begin{aligned} \tilde{G}(p,q) = \tilde{G}(q,p) \qquad \text { for } p \ne q, \end{aligned}$$

(A.3)

and satisfies the distributional identity

$$\begin{aligned} \Delta _p \tilde{G}(p,q) = \delta _q(p) - {\text {area}}(X)^{-1}, \end{aligned}$$

(A.4)

that is,

$$\begin{aligned} \int \tilde{G}(p,q) \Delta f(p) dA(p) = f(q) - {\text {area}}(X)^{-1} \int f dA \end{aligned}$$

for all $C^2$ functions f. Moreover, (A.3) and (A.4) define $\tilde{G}$ uniquely up to an additive constant.

Furthermore, it has the following local behavior: if z is a local coordinate around a point $p_0 \in X$, then

$$\begin{aligned} \tilde{G}(p,q) = -\frac{1}{2 \pi } \log |z(p) - z(q)| + O(1) \end{aligned}$$

(A.5)

uniformly for p and q in a neighborhood of $p_0$, as follows e.g. from the proof of Theorem 4.13(c) in [1] combined with (A.2).

It should be noted that $\tilde{G}$ does not have any special behavior at $p_\infty$. From $\tilde{G}$ we obtain the bipolar Green’s function with pole at $p_\infty$ as follows.

Proposition A.2

The function defined by

$$\begin{aligned} G(p,q) = 2 \pi [\tilde{G}(p,q) - \tilde{G}(p,p_\infty ) - \tilde{G}(q,p_\infty )] \end{aligned}$$

(A.6)

satisfies the properties stated in Proposition 2.1.

Proof

Parts (b) and (c) of Proposition 2.1 follow directly from (A.5) and (A.6) and part (d) follows from (A.6) and the symmetry (A.3) of $\tilde{G}$. Hence it remains to check part (a).

Fix a $q \in X \setminus \{p_\infty \}$. The function $p \mapsto G(p,q)$ is clearly real-valued on $X \setminus \{ p_{\infty }, q\}$. Moreover, it follows from (A.4) and (A.6) that

$$\begin{aligned} \Delta _p G(p,q) = 2 \pi (\delta _q(p) - \delta _{p_\infty }(p)) \end{aligned}$$

in a distributional sense. The right-hand side is zero for $p \notin \{p_\infty ,q\}$, hence $p \mapsto G(p,q)$ is weakly harmonic on $X {\setminus } \{p_\infty ,q\}$. By Weyl’s lemma (see e.g. [38, Theorem 3.4.2]), the function $p \mapsto G(p,q)$ is then harmonic on $X \setminus \{p_\infty , q\}$. This concludes the proof. $\quad \square$