The Existence of Generalized Isothermal Coordinates for Higher Dimensional Riemannian Manifolds

We shall show that, for any given point p on a Riemannian manifold (M, g0), there is a pointwise conformal metric g = -Dg0 in which the g-geodesic sphere centered at p with radius r has constant mean curvature 1 /r for all sufficiently small r . Furthermore, the exponential map of g at p is a measure preserving map in a small ball around p . INTRODUCTION This note is devoted to studying the mean curvature of the geodesic sphere. In particular, we are interested in the existence of geodesic spheres with constant mean curvature under a certain conformal deformation of metrics. In what follows, we always assume that Mn is an n-dimensional C?? smooth manifold and g0 is an arbitrary Riemannian metric on M, simply denoted by (Mn, g?), and we also assume that CD is a strictly positive function. The goal of this paper is to prove the following. Main Theorem. For any given point p on a given Coo (C a resp.) Riemannian manifold (M, g ), there is a local C? (C k-3 a resp.) pointwise conformal metric g = (Dg0 such that the geodesic sphere centered at p with radius r in the metric g has constant mean curvature xactly 11/ for all sufficiently small r. To illustrate the usefulness of the main theorem, we present some applications. First, the exponential map of the metric g given in the main theorem has some nice properties. Let {y1, ... , Yn } be g-geodesic normal coordinates at p and let det(gij(y)) be the determinant of the metric g with respect to {y} . Using the main theorem, one can prove the following corollary (cf. ? 1). Corollary 0.1. Let (M, g0) be a C?? Riemannian manifold and let p be a point on M. Then there exists a conformal metric g = eDg0 such that det gij(y) = 1 Received by the editors January 10, 1989 and, in revised form, April 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10, 35J60; Secondary 53A30, 58G30. () 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page