New Counterexamples to Min-Oo's Conjecture via Tunnels

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.