Connected sum of manifolds with spectral Ricci lower bounds

TL;DR

The paper proves that the connected sum operation preserves a global spectral Ricci lower bound when the ambient dimension satisfies $n\ge 3$ and the coupling parameter is in the supercritical regime $\gamma>\frac{n-1}{n-2}$. The main technique is a Gromov–Lawson–type tunnel construction that glues two manifolds while maintaining a lower bound for $\lambda_1(-\gamma\Delta+\mathrm{Ric})$ up to an arbitrarily small loss, achieved by merging a Green’s function with a convex warped-product tunnel. Key technical contributions include delicate regional estimates of the modified metric and Ricci curvature, and a detailed asymptotic analysis of the Green’s function and metric near the gluing region. The results yield immediate corollaries for connected sums with multiple factors and reveal the sharpness of the threshold, contributing to the understanding of spectral Ricci-type bounds under surgery and to related topological consequences such as Betti-number bounds.

Abstract

Let $n > 2$, $γ> \frac{n-1}{n-2}$, and $λ\in \mathbb{R}$. We prove that if $M$ and $N$ are two smooth $n$-manifolds that admit a complete Riemannian metric satisfying \[ -γΔ+ \mathrm{Ric} > λ, \] then the connected sum $M \# N$ also admits such a metric. The construction geometrically resembles a Gromov-Lawson tunnel; the range $ γ> \frac{n-1}{n-2} $ is sharp for this to hold.

Theorems & Definitions (23)

• Theorem 1
• Corollary 1
• Corollary 2
• Lemma 1
• Proposition 1
• proof : Proof of \ref{['thm:tunnel']}
• Lemma 2
• proof
• Lemma 3
• proof