Research

Abstract: The rigidity theorems of Llarull and Marques–Neves, 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. 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 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. 

Abstract: Min-Oo’s Conjecture is a positive curvature version of the positive mass theorem. Brendle, Marques, and Neves [Invent. Math. 185 (2011), pp. 175–197] produced a perturbative counterexample to this conjecture. In 2021, Carlotto [Living Rev. Relativ. 24 (2021), Article 2] 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. 

Abstract: In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the validity of this conjecture under the additional assumption of rotational symmetry. Furthermore, we obtain a rigidity theorem in dimensions at least three for rotationally symmetric manifolds, which is analogous to the width rigidity theorem of Marques and Neves. We also prove a volume preserving intrinsic flat stability result for this rigidity theorem. Lastly, we study variants of Marques and Neves’ stability conjecture. In the first, we show Gromov–Hausdorff convergence outside of certain “bad” sets. In the second, we assume non-negative Ricci curvature and show Gromov–Hausdorff stability. 

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Abstract: In 2014, Gromov asked if nonnegative scalar curvature is preserved under intrinsic flat convergence. Here we construct a sequence of closed oriented Riemannian 

n-manifolds, n≥3, with positive scalar curvature such that their intrinsic flat limit is a Riemannian manifold with negative scalar curvature. 

Abstract: Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that is the interior of a sufficiently connected, compact, contractible 5-manifold with boundary and supports a complete Riemannian metric with uniformly positive scalar curvature is diffeomorphic to Euclidean 5-space. Next, we investigate the analogous question for compact manifolds with boundary: Must a compact, contractible manifold that supports a Riemannian metric with positive scalar curvature and mean convex boundary necessarily be the disk? We present examples demonstrating that this curvature condition alone cannot distinguish the disk; on the other hand, we exhibit stronger curvature conditions that allow us to draw such a conclusion.