The isoperimetric profile of a smooth Riemannian manifold for small volumes

We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C2,α-close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.

• Publication year

  2007

• Venue

  Annals of Global Analysis and Geometry

• Publication date

  2007-10-07

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1007/s10455-008-9152-6arXiv 0710.1396
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Constant mean curvature spheres in Riemannian manifolds

  2009influential reference
• Calibrations and isoperimetric profiles

  2007cited by this paper
• Riemannian Geometry: Isoperimetric Inequalities (Constant Curvature)

  2006cited by this paper
• The Isoperimetric Problem

  2005cited by this paper
• Constant mean curvature hypersurfaces condensing along a submanifold

  2004cited by this paper
• The periodic isoperimetric problem

  2003cited by this paper
• Houston Journal of Mathematics

  2003cited by this paper
• Isoperimetric sets on Carnot groups

  2003cited by this paper
• Propriétés de concavité du profil isopérimétrique et applications

  2003cited by this paper
• Sharp local isoperimetric inequalities involving the scalar curvature

  2002cited by this paper
• Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces

  2001cited by this paper
• The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

  2000cited by this paper
• Some Sharp Isoperimetric Theorems for Riemannian Manifolds

  2000cited by this paper
• A new isoperimetric comparison theorem for surfaces of variable curvature

  1996cited by this paper
• Riemannian Geometry: A Modern Introduction

  1994cited by this paper
• Stable constant mean curvature tori and the isoperimetric problem in three space forms

  1992cited by this paper
• Foliation by constant mean curvature spheres

  1991influential reference
• Geometric Measure Theory: A Beginner's Guide

  1988cited by this paper
• Partial Differential Relations

  1986cited by this paper
• Minimal surfaces and functions of bounded variation

  1977cited by this paper
• Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints

  1975cited by this paper
• On the first variation of a varifold

  1972cited by this paper

• From bubbles to clusters: Multiple solutions to the Allen–Cahn system

  2026cites this paper
• Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature

  2026influential citation
• Connectedness properties of small minimal clusters in Riemannian or Finsler manifolds

  2025cites this paper
• Isoperimetric Regions in Anisotropically Scaled Product Manifolds

  2025cites this paper
• Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

  2025influential citation
• From bubbles to clusters: Multiple solutions to the Allen--Cahn system

  2024cites this paper
• Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type

  2024cites this paper
• Multiplicity results for mass constrained Allen–Cahn equations on Riemannian manifolds with boundary

  2024influential citation
• Local minimizers of the anisotropic isoperimetric problem on closed manifolds

  2023cites this paper
• Quantitative $C^1$-stability of spheres in rank one symmetric spaces of non-compact type

  2023cites this paper
• Multiplicity of solutions to the multiphasic Allen–Cahn–Hilliard system with a small volume constraint on closed parallelizable manifolds

  2022cites this paper
• Concentration of CMC Surfaces in a 3-manifold

  2022cites this paper
• Double bubbles with high constant mean curvatures in Riemannian manifolds

  2021cites this paper
• On Clusters and the Multi-isoperimetric Profile in Riemannian Manifolds with Bounded Geometry

  2020cites this paper
• Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space

  2020cites this paper
• Lusternik–Schnirelman and Morse Theory for the Van der Waals–Cahn–Hilliard equation with volume constraint

  2020cites this paper
• Nondegeneracy of standard double bubbles

  2019cites this paper
• Isoperimetry and volume preserving stability in real projective spaces

  2019cites this paper
• Existence of CMC-foliations in asymptotically cuspidal manifolds

  2018cites this paper
• On critical points of the relative fractional perimeter

  2018cites this paper
• Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

  2018cites this paper
• Existence of hypersurfaces with prescribed mean curvature I – generic min-max

  2018cites this paper
• Regularity of Isoperimetric Regions that are Close to a Smooth Manifold

  2018cites this paper
• Geometric Variational Problems for Mean Curvature

  2018cites this paper
• On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds

  2017cites this paper
• The isoperimetric problem for lens spaces

  2017cites this paper
• Regularity of Isoperimetric Regions that are Close to a Smooth Manifold

  2017cites this paper
• Min–max theory for constant mean curvature hypersurfaces

  2017cites this paper
• On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

  2017cites this paper
• Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

  2017cites this paper
• Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

  2016cites this paper
• Construction de surfaces à courbure moyenne constante et surfaces minimales par des méthodes perturbatives

  2016influential citation
• Rigidity for critical points in the Lévy-Gromov inequality

  2016cites this paper
• Eternal forced mean curvature flows II: Existence

  2015cites this paper
• The isoperimetric problem of a complete Riemannian manifold with a finite number of $C^0$‑asymptotically Schwarzschild ends

  2015cites this paper
• New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

  2014cites this paper
• Isoperimetric inequalities in convex bodies

  2014cites this paper
• Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

  2013cites this paper
• Concentration of small Willmore spheres in Riemannian 3-manifolds

  2013cites this paper
• Isoperimetric inequalities in Euclidean convex bodies

  2013cites this paper
• Extremal domains for the first eigenvalue in a general Riemannian manifold

  2013cites this paper
• Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile

  2012cites this paper
• Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

  2012cites this paper
• Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

  2012cites this paper
• Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

  2011cites this paper
• Large isoperimetric surfaces in initial data sets

  2011cites this paper
• STABILITY RESULTS FOR SHARP INEQUALITIES AND THEIR APPLICATION IN MATHEMATICAL PHYSICS: An introduction to the critical droplet problem as a case study

  2011cites this paper
• ISOPERIMETRIC INEQUALITIES ON NONPOSITIVELY CURVED SPACES

  2010cites this paper
• The isoperimetric profile of a noncompact Riemannian manifold for small volumes

  2010influential citation
• Area-minimizing regions with small volume in Riemannian manifolds with boundary

  2009influential citation
• Semianalyticity of isoperimetric profiles

  2009cites this paper
• Domaines extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami

  2009cites this paper
• Some local eigenvalue estimates involving curvatures

  2009influential citation
• Small Surfaces of Willmore Type in Riemannian Manifolds

  2009cites this paper
• Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

  2007cites this paper