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Width Stability of Rotationally Symmetric Metrics.

Hunter Stufflebeam1, Paul Sweeney2

  • 1Department of Mathematics, University of Pennsylvania, Philadelphia, PA USA.

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PubMed
Summary
This summary is machine-generated.

This study validates the Marques-Neves conjecture for rotationally symmetric manifolds, proving a new rigidity theorem and stability results in higher dimensions. These findings advance understanding of geometric stability in mathematics.

Keywords:
Convergence of manifoldsScalar curvatureStability

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Area of Science:

  • Differential Geometry
  • Geometric Analysis
  • Topology

Background:

  • The Marques-Neves conjecture addresses volume preserving intrinsic flat stability for the unit round 3-sphere.
  • Their work is based on a width rigidity theorem for the unit round 3-sphere.

Purpose of the Study:

  • To establish the validity of the Marques-Neves conjecture under rotational symmetry.
  • To derive a new rigidity theorem for rotationally symmetric manifolds in dimensions >= 3.
  • To prove volume preserving intrinsic flat stability for the new rigidity theorem.

Main Methods:

  • Analysis of geometric properties of manifolds with rotational symmetry.
  • Application of techniques from geometric analysis and topology.
  • Investigation of Gromov-Hausdorff convergence and stability.

Main Results:

  • The Marques-Neves conjecture is proven valid for rotationally symmetric cases.
  • A novel rigidity theorem analogous to the Marques-Neves width rigidity theorem is established for dimensions >= 3.
  • Volume preserving intrinsic flat stability is demonstrated for the new rigidity theorem.
  • Gromov-Hausdorff convergence and stability are shown for variants of the conjecture, including cases with non-negative Ricci curvature.

Conclusions:

  • The study confirms a significant conjecture in geometric analysis under specific symmetry conditions.
  • New rigidity and stability results are presented for rotationally symmetric manifolds.
  • The findings contribute to a deeper understanding of geometric stability and convergence properties in Riemannian geometry.