Zhu, Bo2022-09-132022-09-132022-06https://hdl.handle.net/11299/241637University of Minnesota Ph.D. dissertation. 2022. Major: Mathematics. Advisors: Jiaping Wang, Tianjun Li. 1 computer file (PDF); 120 pages.The study of the interplay of geometry, topology, and curvature lower bound is animportant topic in differential geometry. Many progresses have been made on the man- ifolds with sectional curvature or Ricci curvature bounded below over the past fifth years ([29, 42, 63]). However, many problems related to the scalar curvature remain conjectural [25, 26, 33, 40, 55, 62, 68] and see the website https://www.spp2026.de/. In this thesis, first, we study the interplay of the geometry and positive scalar cur- vature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature, and a width estimate. In general dimensional manifold, we obtain a volume growth of a geodesic ball. Next, we study the geometry of the mean convex domain in Rn. Then, we prove that for every three-dimensional Riemannian manifold with non-negative Ricci curvature and strictly mean convex boundary, there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary, which answered a question raised by Gromov for three-dimensional case in [31]. Finally, we extend a comparison theorem of minimal Green functions in [52] to harmonic functions on complete non-compact three-dimensional manifolds with compact connected boundary. This yields an upper bound on the integral related to the scalar curvature on complete, non-parabolic three-dimensional manifolds.enThesis or Dissertation