Dependance of Dirichlet integrals upon lumps of Riemann surfaces

Abstract

Take a simple arc $\gamma$ in an open Riemann suface $R$ carrying a nonconstant harmonic function $u$ with finite Dirichlet integral $D(u;R)$. Form a Riemann surface $R_{\gamma}$ with lump $\widehat{\mathbf{C}}\setminus\gamma$ by pasting $R\setminus\gamma$ with $\widehat{\mathbf{C}}\setminus\gamma$ crosswise along $\gamma$, i.e. $R_{\gamma}:=(R\setminus\gamma) \utimes{\gamma}(\widehat{\mathbf{C}}\setminus\gamma)$, and the transplant $u_{\gamma}$ of $u$ on $R$ to $R_{\gamma}$ characterized by its being harmonic on $R_{\gamma}$ with $D(u_{\gamma};R_{\gamma})<+\infty$ and $u_{\gamma}=u$ at the ideal boundary of $R_{\gamma}$ and hence of $R$ in a suitable sense. We are interested in the comparison of $D(u_{\gamma};R_{\gamma})$ with $D(u;R)$ when we take a variety of choices of pasting arcs $\gamma$ in $R$, and we will prove that $D(u_{\gamma};R_{\gamma})<D(u;R)$ for any $u$ level arc $\gamma$ in $R$, $D(u_{\gamma};R_{\gamma})>D(u;R)$ for any $u$ conjugate level arc $\gamma$ in $R$, and as a consequence of these two facts there is a nondegenerate arc $\gamma$ (i.e. not a point arc $\gamma$) in $R$ such that $D(u_{\gamma};R_{\gamma})=D(u;R)$.