The Brownian loop measure on Riemann surfaces and applications to length spectra

Lawler and Werner introduced the Brownian loop measure on the Riemann sphere in studying Schramm-Loewner evolution. It is a sigma-finite measure on Brownian-type loops, which satisfies conformal invariance and restriction property. We study its generalization on a Riemannian surface $(X,g)$. In particular, we express its total mass in every free homotopy class of closed loops on $X$ as a simple function of the length of the geodesic in the homotopy class for the constant curvature metric conformal to $g$. This identity provides a new tool for studying Riemann surfaces' length spectrum. One of the applications is a surprising identity between the length spectra of a compact surface and that of the same surface with an arbitrary number of cusps. This is a joint work with Yuhao Xue (IHES).