Nonlocal phase transitions in codimension-one: diffuse approximation, stability, and asymptotics

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This thesis explores the theory of the fractional perimeter on closed Riemannian manifolds, with a focus on codimension-one phenomena and their approximation through diffuse interface models. A central theme is the existence and regularity of nonlocal minimal surfaces, which arise as critical points of the fractional perimeter, on closed manifolds. Despite their intrinsic nonlocality, which is an anomalous feature compared with the classical theory of minimal surfaces, these surfaces share many structural elements with classical minimal hypersurfaces. In several respects, particularly in the context of stability and finite Morse index, we aim to show that they exhibit improved compactness and regularity properties. These properties make nonlocal minimal surfaces highly suitable for min-max constructions since Morse theory for nonlocal minimal surfaces is, in some sense, as flawless as finite-dimensional Morse theory. At the core of our existence result is the use of the fractional Allen-Cahn equation as a diffuse approximation of nonlocal minimal surfaces. This method has proved to be particularly effective in the context of min-max constructions, as it allows for the existence of many critical points with precise index and energy bounds. Addressing the regularity theory of nonlocal minimal surfaces on manifolds requires several key ingredients: precise local estimates on the heat kernel of complete manifolds, rigidity results for stationary cones stable in $mathbb{R}^n setminus {0}$, and the development of the Caffarelli-Silvestre extension theory on closed manifolds. In this work, we develop (some of) these ingredients and utilize these technical tools to address different problems. For example, the local estimates for the heat kernel will be used both to deduce regularity, via a blow-up procedure, of finite Morse index nonlocal minimal surfaces arising as limits of the fractional Allen-Cahn equation and to characterize the asymptotics of the fractional Laplacian on noncompact manifolds as $sto 0$.