2019/03/28 by Alessandro Carlotto, Chao Li, Carlotto, Alessandro +1 · 1 citation
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1903.11772
openalex publication_date 2019/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a series of results concerning the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. In particular, we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. The methods we employ are flexible enough to allow the construction of continuous paths of positive scalar curvature metrics with minimal boundary, and to derive similar conclusions in that context as well. Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case.