Communications in Analysis and Geometry

Volume 26 (2018)

Number 4

Optimal estimate for the gradient of Green’s function on degenerating surfaces and applications

Pages: 887 – 913



Paul Laurain (Institut de Mathématiques de Jussieu, Paris Rive Gauche, Paris, France)

Tristan Rivière (Department of Mathematics, ETHZ Zurich, Switzerland)


In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with $L^2$-bounded second fundamental form and for Riemannian surfaces of uniformly bounded Gaussian curvature entropy.

Received 8 August 2014

Published 6 September 2018