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Volker Branding1, Georges Habib2,3

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

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Summary
This summary is machine-generated.

This study presents new eigenvalue estimates for the Hodge Laplacian on weighted Riemannian manifolds, unifying existing results and offering geometric insights. An inequality connects Jacobi operator eigenvalues for f-minimal hypersurfaces with the Hodge Laplacian spectrum.

Keywords:
Hodge LaplacianJacobi operatoreigenvalue estimatesweighted manifold

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Area of Science:

  • Differential Geometry
  • Analysis on Manifolds
  • Mathematical Physics

Background:

  • The Hodge Laplacian is a key operator in the analysis of differential forms on manifolds.
  • Weighted Riemannian manifolds and their properties are crucial in various geometric contexts.
  • Understanding eigenvalue estimates is fundamental for spectral geometry.

Purpose of the Study:

  • To derive novel eigenvalue estimates for the Hodge Laplacian on weighted Riemannian manifolds.
  • To unify and extend existing results in the field.
  • To explore geometric applications of these new estimates.

Main Methods:

  • Development of analytical techniques for eigenvalue estimation.
  • Application of differential geometry principles to weighted manifolds.
  • Spectral analysis of the Hodge Laplacian.

Main Results:

  • A unified set of eigenvalue estimates for the Hodge Laplacian.
  • New insights into the spectral properties of differential forms on weighted manifolds.
  • A specific inequality relating Jacobi operator eigenvalues and Hodge Laplacian spectrum for f-minimal hypersurfaces.

Conclusions:

  • The derived estimates provide a significant advancement in spectral geometry.
  • The results offer powerful tools for studying geometric properties of manifolds.
  • The connection to f-minimal hypersurfaces opens new avenues for research.