Research

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 answer this question in the affirmative. 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. 

Abstract: The rigidity theorems of Marques-Neves and of Llarull, which show two different ways scalar curvature can characterize the sphere, have associated stability conjectures. Here we produce the first examples related to these stability conjectures. The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence. The second set of examples constructs sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume preserving intrinsic flat sense. In order to construct such sequences, we improve the Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount. Moreover, we are able to generalize both the sewing construction of Basilio, Dodziuk, and Sormani, and the construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit with no geodesics.