New Counterexamples to Min-Oo's Conjecture via Tunnels

TL;DR

This paper develops a new non-perturbative gluing method to construct counterexamples to Min-Oo's Conjecture, advancing understanding of positive curvature and topology in geometric analysis.

Contribution

It introduces a novel gluing technique based on Gromov--Lawson surgery to produce non-perturbative counterexamples with complex topology for Min-Oo's Conjecture.

Findings

Constructed non-perturbative counterexamples

Produced examples with more complex topology

Extended the scope of counterexamples beyond previous perturbative ones

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.