Connected sum of manifolds with spectral Ricci lower bounds

TL;DR

This paper demonstrates that the connected sum of two manifolds with spectral Ricci lower bounds also admits a metric satisfying similar bounds, extending geometric analysis techniques to manifold constructions.

Contribution

It proves the preservation of spectral Ricci lower bounds under connected sum operations for a specific range of parameters, with a geometric construction resembling a Gromov-Lawson tunnel.

Findings

Connected sums preserve spectral Ricci bounds for b3 > (n-1)/(n-2).

Construction uses a Gromov-Lawson tunnel approach.

Range b3 > (n-1)/(n-2) is sharp.

Abstract

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