# Loss of initial data under limits of Ricci flows

## TL;DR

This paper constructs Ricci flows on a torus that start close to a flat metric but converge to a scaled version, demonstrating loss of initial data in the flow limits.

## Contribution

It provides an explicit example of Ricci flows where initial convergence does not imply convergence of the flows themselves, highlighting limitations in initial data preservation.

## Key findings

- Flows converge to a scaled flat metric, not the initial metric.
- Initial metrics converge in Gromov-Hausdorff sense, but flows do not.
- Demonstrates loss of initial data in Ricci flow limits.

## Abstract

We construct a sequence of smooth Ricci flows on $T^2$, with standard uniform $C/t$ curvature decay, and with initial metrics converging to the standard flat unit-area square torus $g_0$ in the Gromov-Hausdorff sense, with the property that the flows themselves converge not to the static Ricci flow $g(t)\equiv g_0$, but to the static Ricci flow $g(t)\equiv 2g_0$ of twice the area.