ESTIMATES FOR WEIGHTED VOLUMES AND APPLICATIONS By ZHONGMIN QIAN 1 [Received 3 March 1995; in revised form 29 April 1996] §1. Introduction and results WE give a simple approach to weighted volume measures on a Reimann- ian manifold. Weighted volume measures arise naturally from the study of confonnal deformation of a Riemannain metric. Let (M, g) be an n -dimensional and complete Riemannian manifold, and let Ag and ft g denote the Laplace-Beltrami operator and the Riemannian volume measure respectively. In a local chart, write g = (g tj ). Then where (g^) = (g//)" 1 and g = det (gq). Suppose the metric g is confonnally deformed by a positive smooth function a on M, that is, let g be a new metric on M defined by g-(X, Y) = (Tg(X, Y), VZ, Y e TM. Then the volume measure fig is the weighted volume measure cr^ 2 d/i g , and the transform formula of Ricci curvature is given by n — 2 n — 2 ic, = Ric, — Hess hi O- + - — V i n a ® Vln o- where the Hessian and gradient are computed using the metric g. We want to establish geometric results relating to the metric g~ by using the data associated to the original metric g. To this end we need a concept of curvature associated to a weighted 1 The research was supported by EPSRC grant GR/J55946. Quart. J. Math. Oxford (2), 48 (1997), 235-242 © 1997 Oxford Univenity Press Downloaded from https://academic.oup.com/qjmath/article-abstract/48/2/235/1550926 by guest on 26 May 2020