Width Stability of Rotationally Symmetric Metrics

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.