Abstract:

The gradients of geodesic-length functions along systolic curves are studied. We show that the $L^p (1\leq p \leq \infty)$-norms of them at every closed hyperbolic surface $X$ are uniformly comparable to (systole(X))^{1/p}. Several applications to the Weil-Petersson geometry are discussed. For examples: (1). we reprove that the square root of the systole function is uniformly Lipschitz on the Teichmuller space endowed with the Weil-Petersson metric; (2). we also show that the minimal Weil-Petersson holomorphic sectional curvature at any closed hyperbolic surface is bounded above by a uniform negative constant independent of genus.