Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

Abstract In this article we show that for every finite area hyperbolic surface X of type {(g,n)} and any harmonic Beltrami differential ? on , then the magnitude at point small injectivity radius is uniform bounded from above by ratio Weil–Petersson norm over square root systole up to a positive constant multiplication. We apply bound Ricci curvature, restricted short in moduli space, uniformly below negative reciprocal As an application, average total scalar curvature space comparable {-g} as genus g goes infinity.