Stability, Finiteness and Dimension Four

TL;DR

This paper proves that in four-dimensional Riemannian geometry, only finitely many manifold types exist under fixed curvature, volume, and diameter constraints, highlighting stability and finiteness properties.

Contribution

It establishes a finiteness theorem for closed 4-manifolds with bounded curvature, volume, and diameter, extending understanding of geometric stability in four dimensions.

Findings

Finiteness of diffeomorphism types under given bounds

Bounded sectional curvature, volume, and diameter imply only finitely many manifold types

Advances the classification of 4-manifolds in Riemannian geometry

Abstract

We prove that for any $k\in \mathbb{R},$ $v>0,$ and $D>0$ there are only finitely many diffeomorphism types of closed Riemannian $4$-manifolds with sectional curvature $\geq k,$ volume $\geq v,$ and diameter $\leq D.$