GENERALIZED ISOPERIMETRIC INEQUALITIES FOR EXTRINSIC BALLS IN MINIMAL SUBMANIFOLDS Steen Markvorsen and Vicente Palmer* Abstract. The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g. [3] and [10]. When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see [11]. This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms. 1. Introduction Let P m be an immersed submanifold of a complete riemannian manifold N n . The distance function on the ambient space N n is denoted by d. If p is a point in P , we define r(q) := d(p, q) for every q ∈ N . We shall also denote by r the restriction r| P : P −→ IR. Let us define the extrinsic ball of radius R and center p ∈ P , D R (p) ⊆ P , to be the smooth connected component of B n R (p) ∩ P = {q ∈ P | r p (q) ≤ R} which contains p. Here, B n R (p) denotes the geodesic R-ball around p in the ambient space N , subject to the restriction that R ≤ min{i N (p), π 2 √ b }, where b is the supremum of the sectional curvatures of N , and i N (p) is the injectivity radius of N from p. We shall denote by S n−1 R (p) the geodesic R-sphere in N . We observe that, when considering the n-dimensional simply connected space forms of constant curvature b ∈ IR, (denoted from now on as IK n (b)) and its totally geodesic submanifolds IK m (b) ⊆ IK n (b), then the corresponding extrinsic R-ball cen- tered at ˜ p ∈ IK m (b) is just the geodesic R-ball B b,m R centered at ˜ p in this submanifold, and its boundary is the geodesic sphere S b,m−1 R . 1991 Mathematics Subject Classification. 53C21; 58J65. Key words and phrases. isoperimetric inequality, Brownian motion, mean exit time, minimal submanifold, extrinsic ball. *Supported by a Grant of the Spanish Ministerio de Educaci´ on y Cultura Typeset by A M S-T E X 1