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Sharp Cheeger-Buser Type Inequalities in Spaces.

Nicolò De Ponti1, Andrea Mondino2

  • 1Dipartimento di Matematica "Casorati", Università degli Studi di Pavia, Pavia, Italy.

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|March 22, 2021
PubMed
Summary
This summary is machine-generated.

This study refines inequalities relating Cheeger's isoperimetric constant (h) and the first eigenvalue of the Laplacian. New dimension-free bounds are established for metric measure spaces with Ricci curvature bounded below.

Keywords:
Buser inequalityCheeger inequalityFirst eigenvalue laplace operatorMetric measure spacesRicci curvature

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Area of Science:

  • Differential Geometry
  • Geometric Analysis
  • Analysis on Metric Spaces

Background:

  • Cheeger's 1970 lower bound relates the first eigenvalue of the Laplacian to the isoperimetric constant for Riemannian manifolds.
  • Buser (1982) and Ledoux (2004) established upper bounds for this relationship, with Ledoux achieving a dimension-free estimate for manifolds with bounded Ricci curvature.
  • Existing bounds primarily apply to smooth Riemannian manifolds.

Purpose of the Study:

  • To sharpen and generalize existing inequalities between Cheeger's isoperimetric constant (h) and the first eigenvalue of the Laplacian.
  • To establish a dimension-free sharp Buser inequality for spaces with Bakry-Émery weighted Ricci curvature bounded below.
  • To extend these results to a broader class of non-smooth metric measure spaces.

Main Methods:

  • Development of novel analytical techniques to refine existing inequalities.
  • Application of concepts from Bakry-Émery calculus for weighted Ricci curvature.
  • Generalization of bounds to the framework of synthetic Ricci curvature in metric measure spaces.

Main Results:

  • A dimension-free sharp Buser inequality is derived for spaces with Bakry-Émery weighted Ricci curvature bounded below, sharp on Gaussian spaces.
  • The established inequalities hold for a more general class of (possibly non-smooth) metric measure spaces.
  • The results provide refined quantitative relationships between spectral-geometric quantities.

Conclusions:

  • The study successfully generalizes and sharpens spectral-geometric inequalities.
  • The findings extend to non-smooth spaces, broadening the applicability of Cheeger-type inequalities.
  • This work offers a more precise understanding of the interplay between curvature and spectral properties in metric measure spaces.