The Anomaly Flow over Riemann Surfaces

Abstract

We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of initial data where the solution exists for all time. A geometric interpretation of these results is given in terms of the Anomaly flow on a Calabi-Yau threefold.

A flow of conformally balanced metrics with Kähler fixed points

“…It also inspired directly the solutions of the Fu-Yau Hessian equations found recently in [62], which extended our earlier work [55,58]. However, it is still largely unexplored, and its long-term behavior is already known to exhibit in general a rather intricate behavior, such as a particular sensitivity to the initial data [20,60,61].…”

New curvature flows in complex geometry

“…Next we describe a very recent work of T. Fei, Z. Huang, and S. Picard [34,35]. Their set-up is certain hyperkähler fibrations over Riemann surfaces originating from works of Fei [31,32,33], which are themselves built on earlier constructions of Calabi [14] and Gray [49].…”

“…where all norms and operators are taken with respect to the evolving metric ω(t). The following criterion for the long-time existence of the flow in this setting was proved in [35]:…”

“…At the other end, it was proved in [35] that for initial data with e f large, then the flow will exist for all time, and collapse the fibers in the limit:…”

“…which is monotone decreasing for arbitrary initial data. This functional was used in [35] to provide convergence of the normalized solution in the regime u ≥ 0. It is possible that the existence of solutions to the Hull-Strominger system may be tied to some stability condition.…”