The Anomaly flow over Riemann surfaces

TL;DR

This paper introduces a new nonlinear parabolic flow on Riemann surfaces derived from the Anomaly flow, providing criteria for long-term existence and finite-time singularity formation, with geometric implications for Calabi-Yau threefolds.

Contribution

It develops the theory of a novel flow on Riemann surfaces, including existence criteria and singularity conditions, linking it to the Anomaly flow on Calabi-Yau threefolds.

Findings

Criteria for long-time existence of the flow

Conditions leading to finite-time singularities

Geometric interpretation related to Calabi-Yau threefolds

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.