Minimal surfaces and the Almgren-Pitts Min-Max theory

• Grant number: 17/05800-6
• Support Opportunities: Scholarships in Brazil - Master
• Start date: June 01, 2017
• End date: August 28, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
• Principal Investigator: Alexandre Paiva Barreto
• Grantee: Anderson Felipe Viveiros
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 12/24454-8 - Algebraic, geometric and differential topology, AP.TEM
• Associated scholarship(s): 18/08511-8 - Min-max minimal surfaces, BE.EP.MS

Abstract

Since the works of Schoen and Yau in the late 70's, it became clear that the existence of a minimizing area surface might to exert great influence on the geometry of a Riemannian 3-manifold. Along the last few years, researchers as M.Cai, G.J.Galloway, H.Bray, S.Brendle, F.C.Marques and A.Neves have proved several rigidity results which made use of the existence of such minimizing surfaces hypothesis. In this master project, we are interested in studying the Almgren-Pitts Min-Max method for construction of minimal surfaces in tridimensional manifolds. Although minimal surfaces obtained by this method do not minimize area, in general, it is natural to ask ourselves if rigidity results could also be obtained for these surfaces.Our main goal is to study this last question and, as starting point, we have chosen the article "Rigidity of Min-Máx minimal spheres in three-manifolds" by F.C.Marques and A.Neves which, besides serving as an excellent "gateway" to the Min-Max method, also presents other important tools as, for instance, the Ricci flows. (AU)