Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

TL;DR

This paper demonstrates that Riemannian manifolds with positive Ricci curvature and specific eigenvalue conditions approximate the product of standard spheres in the Gromov-Hausdorff sense.

Contribution

It establishes a new link between eigenvalue pinching conditions and geometric convergence to sphere products.

Findings

Manifolds with positive Ricci curvature approximate sphere products.

Eigenvalue conditions imply Gromov-Hausdorff convergence.

Provides new insights into geometric stability under spectral constraints.

Abstract

We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.