Geometric Variational Problems in Metric Measure Spaces: Area, Perimeter, and Regularity

This paper explores geometric variational problems, specifically those involving area and perimeter minimization, within the general framework of metric measure spaces (MMS). We delve into the foundational concepts of perimeter and area in these non-Euclidean settings, addressing the challenges posed by the lack of a smooth structure. The study reviews established methods for defining and analyzing these quantities, drawing connections to rectifiability and generalized notions of curvature. A significant portion is dedicated to investigating the regularity properties of minimizers, particularly focusing on how conditions within the MMS, such as doubling measures and Poincaré inequalities, influence the smoothness and structure of solutions. The paper synthesizes classical results with recent advancements, highlighting the profound implications for areas like optimal transport, analysis on singular spaces, and the theory of spaces with lower Ricci curvature bounds.