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Symmetrization procedures for the isoperimetric problem in symmetric spaces of noncompact type

Daniel John

TL;DR

This paper develops a Steiner-type symmetrization for domains in symmetric spaces of noncompact type using 1-parameter transvections. The core idea is to reduce the isoperimetric problem to a variational problem for a height function on the orbit space, establishing the existence, uniqueness, and smoothness of a volume-preserving, area-minimizing symmetrized domain via a carefully analyzed area functional and its Euler–Lagrange equation. It provides gradient estimates and a robust approximation scheme to ensure regular minimizers, while also showing that convexity of isoperimetric regions is not automatic in this general setting. In the complex hyperbolic case, the method yields explicit computations and suggests that balls may be the isoperimetric minimizers, illustrating the potential applicability to broader geometric contexts.

Abstract

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.

Symmetrization procedures for the isoperimetric problem in symmetric spaces of noncompact type

TL;DR

This paper develops a Steiner-type symmetrization for domains in symmetric spaces of noncompact type using 1-parameter transvections. The core idea is to reduce the isoperimetric problem to a variational problem for a height function on the orbit space, establishing the existence, uniqueness, and smoothness of a volume-preserving, area-minimizing symmetrized domain via a carefully analyzed area functional and its Euler–Lagrange equation. It provides gradient estimates and a robust approximation scheme to ensure regular minimizers, while also showing that convexity of isoperimetric regions is not automatic in this general setting. In the complex hyperbolic case, the method yields explicit computations and suggests that balls may be the isoperimetric minimizers, illustrating the potential applicability to broader geometric contexts.

Abstract

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.

Paper Structure

This paper contains 17 sections, 28 theorems, 120 equations, 1 figure.

Table of Contents

  1. Main Results
  2. Symmetrization in symmetric spaces
  3. Symmetric spaces and transvections
  4. Surface area calculations
  5. Construction of the symmetrization procedure
  6. Analytic properties of the variational problem
  7. The area functional
  8. The Euler-Lagrange equation
  9. Approximating minimizing problems
  10. Alternative symmetrization procedures
  11. The gradient estimate
  12. Differential equation for the gradient
  13. Preliminary estimates
  14. The final estimates
  15. Application to the isoperimetric problem
  16. ...and 2 more sections

Key Result

Theorem 1

Let \widehat{M}^n = \mathsf{G} / \mathsf{K} be a symmetric space of noncompact type. Consider a regular domain \widehat{\Omega} \subset\subset \widehat{M}^n and a transvection \tau such that the following holds: Then the symmetrization procedure of Definition 59 assigns to \widehat{\Omega} a unique symmetrized domain S(\widehat{\Omega}) \subset \widehat{M}^n of equal volume but smaller (or equal)

Figures (1)

  • Figure 1: Symmetrization using transvections

Theorems & Definitions (73)

  • Definition 1: Symmetrization
  • Theorem 1: Symmetrization
  • Remark 1.1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 63 more