Smoothing and non-smoothing via Ricci Flow
Final Report Abstract
The project was devoted to construct smooth and non-smooth solutions of geometric flows. It is well-known that for the Ricci flow with a smooth initial condition in dimension n > 2, finite time singularity formation is somehow a generic property. So far, this phenomenon has not been observed in two dimensions, where the Ricci flow provides smooth solutions even for H 2 initial conditions. The question therefore arose whether we—based on results for the harmonic map heat flow—can also construct two-dimensional non-smooth Ricci flow solutions. If we extend the question to the flow of prescribed curvature, methods have emerged to solve open problems (the smoothing conjectures) in metric geometry. The latter was also the key question in our work. By modifying the prescribed Gaussian curvature flow on a closed Riemannian surface (M, g¯ ) with negative Euler characteristic, we were able to find conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures fλ = f + λ are given as the sum of a prescribed function f ∈ C ∞ (M ) and an additive constant λ. In contrast to previous work, our approach does not require any sign conditions on f . Moreover, we exhibit conditions under which the function fλ is sign changing and the standard prescribed Gauss curvature flow is not applicable. Furthermore, we searched for smooth solutions of the curvature flow—i.e. conformal, Riemannian metrics—on compact Riemannian surfaces which have prescribed Gaussian curvature in the interior and vanishing geodesic curvature on the boundary ∂M . This generalises a result of Simon Brendle, in which he examines the analogous question for constant Gaussian curvature (i.e. the Ricci flow) in the interior.
Publications
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Lights Out on graphs. Mathematische Semesterberichte, 68(2), 237-255.
Berman, Abraham; Borer, Franziska & Hungerbühler, Norbert
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A variant prescribed curvature flow on closed surfaces with negative Euler characteristic. Calculus of Variations and Partial Differential Equations, 62(9).
Borer, Franziska; Elbau, Peter & Weth, Tobias
