The primary goal of this project was to develop a framework for studying smooth metric measure spaces using techniques from conformal geometry, with the specific aim of improving our understanding of the relationship between the total scalar curvature functional and Perelman's entropy functional. Smooth metric measure spaces arise as a natural setting to study diffusion processes on smooth manifolds, including the Ricci flow. The Ricci flow was famously used by Hamilton and Perelman to resolve the Poincaré conjecture in dimension three. A key ingredient in that work was Perelman's entropy functional, which provides a monotone quantity along the Ricci flow useful for ruling out certain types of collapsing. On the other hand, conformal geometry provides a close relationship between geometric invariants on a Riemannian manifold and sharp Sobolev inequalities which, for instance, provides geometric criteria to rule out certain types of collapse in that setting. These goals and more were achieved through a series of works by the PI. A general framework for studying and understanding conformal geometry on smooth metric measure spaces was formulated, including a robust method for constructing most conformal invariants in this setting. In particular, this work provides new geometric obstructions to the existence of special geometric structures on a given Riemannian manifold which are formally analogous to Einstein metrics. Another application of this framework is a new realization of conformally covariant fractional powers of the Laplacian on the boundary of a Poincaré--Einstein manifold in terms of conformally covariant weighted operators in the interior. In particular, this insight allowed the PI and S.-Y. A. Chang to give geometric criteria for the boundary operators to be positive and satisfy a strong maximum principle. A one-parameter family of curvature functionals was identified which interpolates between the total scalar curvature functional and Perelman's entropy functional. It was shown that this family of functionals possesses the same geometric, analytic, and topological properties as its endpoint cases. Indeed, it is closely related to a family of sharp Gagliardo--Nirenberg inequalities. This proposal also included the development, joint with P. Yang, of new tools to study three-dimensional CR manifolds. These are the abstract analogues of boundaries of domains in C2. The new tools developed are a conformally covariant differential operator acting on CR pluriharmonic functions and an associated scalar invariant whose integral is a CR invariant which, in the case of boundaries of domains, helps control the topology of the domain. The results of this project have many immediate applications in physics and are expected to find wide applications in the other sciences: A special case of the Einstein-like structures on smooth metric measure spaces are static metrics. The obstructions obtained for the former spaces apply to give a simple test to check which spaces can arise as static slices in vacuum solutions to Einstein equations in general relativity. The study of Poincaré--Einstein manifolds is intimately connected to the AdS/CFT correspondence in string theory. On the other hand, nonlocal operators have many applications in financial mathematics and the modeling of physical phenomena. The techniques developed in this project to study conformally covariant fractional powers of the Laplacian on boundaries of Poincaré--Einstein manifolds have direct applications to questions related to the AdS/CFT correspondence and are expected to find eventual applications in the study of more general nonlocal operators. The sharp logarithmic Sobolev inequality plays an important role in many questions related to probability theory. Related Gagliardo--Nirenberg inequalities play an important role in controlling the longtime behavior of the fast diffusion equation, which is used in modeling various physical diffusion processes. The established relationship between the total scalar curvature functional and Perelman's entropy functional has also led the PI to develop a program to establish fully nonlinear and/or higher order analogues of the sharp logarithmic Sobolev inequality and similar inequalities. These are expected to have many physical and probabilistic applications.
