Annals of Global Analysis and Geometry 26: 333–354, 2004. C  2004 Kluwer Academic Publishers. Printed in the Netherlands. 333 The Isoperimetric Problem in Spherical Cylinders RENATO H. L. PEDROSA Departamento de Matem ´ atica - IMECC, Universidade Estadual de Campinas, CP 6065, 13083–970 Campinas, Brazil. e-mail: pedrosa@ime.unicamp.br (Received: 29 September 2003; accepted: 20 July 2004) Abstract. The classical isoperimetric problem for volumes is solved in R × S n (1). Minimizers are shown to be invariant under the group O(n) acting standardly on S n , via a symmetrization argument, and are then classified. Solutions are found among two (one-parameter) families: balls and sections of the form [a, b] × S n . It is shown that the minimizers may be of both types. For n = 2, it is shown that the transition between the two families occurs exactly once. Some results for general n are also presented. Mathematics Subject Classifications (2000): primary 53C42; secondary 53A10. Key words: isoperimetric problem, symmetrization, constant mean curvature submanifolds. 1. Introduction and Statement of Results Let ( M n , g) be a complete Riemannian manifold. The Hausdorff measures on M will be denoted by H k or vol k . An H n -measurable set will be called simply mea- surable. The isoperimetric profile function P : [0, vol n ( M )) → R + is P (v) = inf{H n−1 (∂):  ⊂ M is compact, smooth, with H n () = v}. If vol n ( M ) < ∞, extend P by P (vol n ( M )) = 0( may fail to be connected in this definition). A solution to the isoperimetric problem in ( M, g) with volume v is a compact subset , called either an isoperimetric region or domain, with H n () = v and H n−1 -measurable boundary ∂ such that H n−1 (∂) = P (v). To avoid inessential subsets one usually requires a solution to be the closure of an open set (justified by the regularity result mentioned below). Results on the isoperimetric profile and on general isoperimetric inequalities may be found in [7, 8, 10, 15, 21–23, 27]. In the past decade, the complete classification of isoperimetric domains has been extended to spaces other than the simply-connected space forms. For the latter, with an ample set of references, see [8]. Examples of work including classification are [4, 13, 26, 28–30]. Surveys of recent results are [31–33]. In this work we study the isoperimetric problem in the Riemannian product R × S n and show that they reflect quite closely the elementary two-dimensional case of the right cylinder R × S 1 . Solutions exist for R × S n , by a result of F. Morgan [20] providing existence of (compact) minimizers if M/ G is compact, where G is the isometry group of ( M, g).