Perimeter minimizing sets in RCD spaces

Thesis

This thesis is about recent developments in the study of perimeter minimizing sets in RCD spaces, based on the works [94, 70] and an unpublished note.

After the introduction presented in Chapter 1, where we motivate the study of perimeter minimizing sets in RCD spaces and give an overview of our contributions, in Chapter 2 we collect the background material relevant to our purposes.

In Chapter 3 we present a monotonicity formula for perimeter minimizers in RCD(0, N) metric measure cones and derive a few applications, most notably a stratification result for the singular set of perimeter minimizing sets in non-collapsed RCD(K, N) spaces. We also show that if (X, d, H^N ) is a non-collapsed RCD(0, N) space with Euclidean volume growth containing an entire perimeter minimizer, then every blow-down of X contains a globally perimeter minimizing cone. This last result is new even in smooth Riemannian geometry. This chapter is based on a joint work with Mondino and Semola [94].

Chapter 4 is based on a joint work with Cucinotta [70]. We prove a regularity result for perimeter minimizing sets in spaces arising as pointed Gromov-Hausdorff limits of Riemannian manifolds with uniform two-sided bounds on the Ricci curvature. More specifically, we show that the Hausdorff dimension of the singular set of perimeter minimizers is at most n − 5, where n is the dimension of the ambient space. We also provide an example showing that our estimate is sharp.

Chapter 5 presents some results regarding Modica-Mortola-type approximations of the isoperimetric problem in non-collapsed RCD metric measure spaces. Specifically, we show how the perimeter functional can be approximated by suitable Cahn-Hilliard functionals [51, 141, 140]. We also explore the case where the approximating functionals are defined on a sequence of pointed Riemannian manifolds converging in the pointed Gromov-Hausdorff sense to a limit space (X, d, p), on which the perimeter functional is defined.

Authors: Fiorani, F

• DOI: 10.5287/ora-eydparvjm
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: geometric analysis; differential geometry; metric geometry
• Subjects: Mathematics