New Counterexamples to Min-Oo's Conjecture via Tunnels
Paul Sweeney
TL;DR
The paper addresses Min-Oo's Conjecture, a positive-curvature analogue of the positive mass theorem, by advancing beyond perturbative counterexamples. It develops a quantitative gluing framework based on Gromov–Lawson–Schoen–Yau tunnels to perform controlled connect-sums with ${\mathbb S}^n_+$ while preserving a scalar curvature lower bound $R_{\tilde{g}} > n(n-1)$. The main result shows that, for any $n \ge 3$ and any manifold $(M^n,g)$ with $R_g > n(n-1)$, one can obtain a metric on $N = M \# {\mathbb S}^n_+$ with $R_{\tilde{g}} > n(n-1)$, boundary data matching the standard ${\mathbb S}^{n-1}$, and total geodesy of the boundary, with the added flexibility to enforce large diameter or volume and to realize nontrivial topologies (e.g., ${\mathbb S}^n_+ \# ({\mathbb S}^p \times {\mathbb S}^q)$). These counterexamples broaden the landscape of Min-Oo-type phenomena and address Carlotto’s gluing question by enabling non-perturbative constructions. The work thus significantly enlarges the repertoire of metrics that violate Min-Oo's Conjecture and highlights the role of tunnel-based gluing in high-curvature geometry.
Abstract
Min-Oo's Conjecture is a positive curvature version of the positive mass theorem. Brendle, Marques, and Neves produced a perturbative counterexample to this conjecture. In 2021, Carlotto asked if it is possible to develop a novel gluing method in the setting of Min-Oo's Conjecture and in doing so produce new counterexamples. Here we build upon the perturbative counterexamples of Brendle--Marques--Neves in order to construct counterexamples that make advances on the theme expressed in Carlotto's question. These new counterexamples are non-perturbative in nature; moreover, we also produce examples with more complicated topology. Our main tool is a quantitative version of Gromov--Lawson Schoen--Yau surgery.