Monotonicity formulas via parabolic-to-elliptic transformations: applications to the Ricci flow and fractional heat operators - Ignacio Bustamante (2025)

This thesis contributes to the development of a unified theory of parabolic-to-elliptic transformations, which interprets parabolic partial differential equations as high-dimensional limits of their elliptic counterparts. Our work advances this framework through two principal contributions. First, we extend this connection to fractional operators, enabling new derivations of monotonicity formula for fractional parabolic equations from known elliptic results. Second, we deepen the geometric understanding of the relationship between Colding’s monotonic volume and Perelman’s entropy functional for the Ricci flow. While Perelman’s reduced volume was previously known to emerge as a high-dimensional limit of the Bishop–Gromov relative volume, the geometric origins of the entropy functional W had remained elusive. We demonstrate that both functionals naturally arise from a unified high-dimensional framework via Perelman’s N-space, providing a complete elliptic foundation for these fundamental parabolic quantities.