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Isoperimetric inequalities and spectral consequences in warped product manifolds

Avas Banerjee

Abstract

In this article, we investigate the centered isoperimetric inequality on Cartan-Hadamard manifolds endowed with a warped product structure, namely, among all bounded measurable sets of finite perimeter and prescribed volume, the geodesic ball centered at the pole minimizes the perimeter. Exploiting the interplay between this inequality and the underlying warped product structure, we derive several necessary geometric conditions, some of which are closely related to and comparable with phenomena identified in the work of Simon Brendle [Publ. Math. Inst. Hautes Études Sci. 117 (2013)]. We also establish a sufficient condition ensuring the validity of the centered isoperimetric inequality in this setting. Furthermore, by introducing a suitable isoperimetric-type quotient, we obtain an improvement of the classical Cheeger inequality for a broad class of manifolds. Finally, we derive a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole, valid for a certain class of Riemannian manifolds.

Isoperimetric inequalities and spectral consequences in warped product manifolds

Abstract

In this article, we investigate the centered isoperimetric inequality on Cartan-Hadamard manifolds endowed with a warped product structure, namely, among all bounded measurable sets of finite perimeter and prescribed volume, the geodesic ball centered at the pole minimizes the perimeter. Exploiting the interplay between this inequality and the underlying warped product structure, we derive several necessary geometric conditions, some of which are closely related to and comparable with phenomena identified in the work of Simon Brendle [Publ. Math. Inst. Hautes Études Sci. 117 (2013)]. We also establish a sufficient condition ensuring the validity of the centered isoperimetric inequality in this setting. Furthermore, by introducing a suitable isoperimetric-type quotient, we obtain an improvement of the classical Cheeger inequality for a broad class of manifolds. Finally, we derive a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole, valid for a certain class of Riemannian manifolds.
Paper Structure (10 sections, 25 theorems, 244 equations)

This paper contains 10 sections, 25 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Functional and Geometric analytic preliminaries
  3. Riemannian warped product manifolds
  4. Various types of curvatures on M
  5. Perimeter on M
  6. Isoperimetric inequalities in manifold
  7. Symmetrization in manifold
  8. Spectrum of the Laplace-Beltrami operator and spherical harmonics
  9. Some Auxiliary results
  10. Proof of main theorems

Key Result

Theorem 1.1

Let $(\mathbb{M}^n, g)$ be an $n$-dimensional Riemannian manifold. Assume that the $L^1$-type Pólya--Szegö inequality holds on $\mathbb{M}^n$, namely for every $u \in W^{1,1}_0(\mathbb{M}^n)$, where $u^\sharp$ denotes the centered symmetric decreasing rearrangement of $u$, defined in (centered symmetrization), and $W_0^{1,1}(\mathbb{M}^n)$ is the completion of the compactly supported smooth funct

Theorems & Definitions (51)

  • Theorem 1.1
  • Definition 1.1: Warped product manifold
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Theorem 1.4
  • Remark 1.3
  • Theorem 1.5
  • Definition 2.1
  • ...and 41 more