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Surgery and positive Bakry-Émery Ricci curvature.

Philipp Reiser1, Francesca Tripaldi2

  • 1Department of Mathematics, University of Fribourg, Fribourg, Switzerland.

Calculus of Variations and Partial Differential Equations
|January 15, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces new theorems preserving weighted Riemannian metrics with positive Bakry-Émery Ricci curvature during manifold surgeries. These local results demonstrate that all closed, simply-connected spin 5-manifolds admit such metrics.

Keywords:
53C2053C2154E3557R65

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Area of Science:

  • Differential Geometry
  • Topology
  • Geometric Analysis

Background:

  • Ricci curvature is a fundamental concept in Riemannian geometry, crucial for understanding manifold structure.
  • Surgery operations are key tools for constructing and classifying manifolds.
  • Weighted Riemannian metrics generalize standard metrics, allowing for broader applications.

Purpose of the Study:

  • To establish theorems for preserving positive Bakry-Émery Ricci curvature under manifold surgery.
  • To develop local conditions for such preservation, contrasting with existing global methods.
  • To apply these findings to construct new examples of manifolds with positive Ricci curvature.

Main Methods:

  • Developing novel techniques for analyzing weighted Riemannian metrics under surgery.
  • Establishing two main theorems: one for connected sums and another for surgeries along higher-dimensional spheres.
  • Utilizing local assumptions on the metric around the surgery site.

Main Results:

  • Two theorems are proven for preserving positive Bakry-Émery Ricci curvature along connected sums and higher-dimensional sphere surgeries.
  • The results are local, requiring conditions only near the surgery sphere.
  • It is shown that all closed, simply-connected spin 5-manifolds admit positive Bakry-Émery Ricci curvature.

Conclusions:

  • The study provides powerful new tools for constructing manifolds with positive curvature.
  • The local nature of the theorems simplifies the application of surgery techniques.
  • New examples of manifolds with positive Ricci curvature are generated, advancing geometric analysis.