Symmetrization Procedures for the Isoperimetric Problem in Symmetric Spaces of Noncompact Type

We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equation of mean curvature type. We conclude the paper discussing possible applications to the isoperimetric problem in symmetric spaces of noncompact type. Introduction In this article we consider the isoperimetric problem in symmetric spaces of noncompact type, i.e., the problem of determining the domains minimizing surface area among all regions with a given volume. As existence and partial regularity of isoperimetric solutions in these spaces are given by geometric measure theory [Mo, pp. 129], the goal here is to get some information about the shape of isoperimetric solutions in these spaces. In the history of the isoperimetric problem symmetrization procedures have been a very important tool. J. Steiner (1838), H. A. Schwarz (1884), and E. Schmidt (1943) used symmetrization arguments to get insight into the behavior of isoperimetric solutions in Rn, Hn, and Sn, finally proving the isoperimetric property of metric balls in constant curvature spaces [BZ]. Beginning in 1989 with the work of W.-T. Hsiang and W.-Y. Hsiang [Hs] the isoperimetric problem has been investigated in spaces like Hn × Rm, H n × Hm, Sn × S1, Rn × S1, Hn × S1, or Sn × R by R. Pedrosa, M. Ritoré, and D. John, [P, PRi, J]. In these manifolds the initial technical tool always is a symmetrization argument reducing the problem to the 2–dimensional quotient of the product space by the isotropy group. In some 3–dimensional space forms, for example RP 3, stability arguments have been applied successfully by M. Ritoré and A. Ros [RiRo, Ro]. Up to now the isoperimetric problem has been investigated only in such special manifolds. Techniques suitable for more general symmetric spaces are largely unknown. Date: February 1, 2008. 2000 Mathematics Subject Classification. Primary 49Q10; Secondary 53C42, 35J60.