Energy and foliations on Riemann surfaces (Q2570839)

Let \(M\) be a closed Riemann surface with a smooth metric \(g\), compatible with the complex structure of \(M\), and having constant curvature \(K\). Let \(\mu\) be the volume form on \(M\) induced by \(g\). Let \(\mathcal{F}\) be a foliation of \(M\) given by a unit vector field \(X\) with isolated singularities at \(a_1,\dots ,a_m\) with respective indices \(n_1,\dots ,n_m\). Denote by \(\Sigma\) the set of all singular points. Let \(k_1,k_2\) be geodesic curvatures of the foliations \(\mathcal{F},\mathcal{F}^\perp\) where \(\mathcal{F}^\perp\) is given by the vector field \(JX\). The author proves the following theorem: The foliation \(\mathcal{F}\) is harmonic if and only if there is a smooth nowhere vanishing parameter \(f\) on \(M\setminus\Sigma\), defined up to a multiplicative constant, such that the vector field \(fX\) is the real part of a meromorphic vector field. The energy integral \(\int_M ( k_1^2+k_2^2)\mu\) diverges and the finite part of the energy is given by \(\int_M K\ln f\mu\). Moreover, \(\int_M K\ln f\mu=-2\pi KC\operatorname{Vol} (M)\) for some constant \(C\) that depends on the Green function \(G\) of \(M\) at the singular points and on the parameter \(f\). If there is a point \(p\) in \(M\) with \(f(p)=1\) then the constant \(C\) can be written as \(C=\sum_{i=1}^m n_i G(a_i,p)\). The author generalizes the above result to principal circle bundles over Riemann surfaces.