Long-Time Behavior of the $\alpha$-Ricci Flow on the 2-Torus

We study the $\alpha$-Ricci flow $(g(t),\alpha (t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus $M$. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the $\alpha$-Ricci flow depends on the cohomology class $[\alpha ]\in H^1(M,\mathbb{R})$, which is invariant under the flow. If $[\alpha ]$ is rational, then as $t\to \mathrm{\infty }$, $(M,t^{-1}g(t))$ Gromov–Hausdorff converges to $S^1$ with length $\mathrm{\ell }([\alpha ])\mathrm{/}\sqrt{2}$ where $\mathrm{\ell }([\alpha ])$ is the primitive length of $[\alpha ]$. On the universal cover, the rescaled flow $(s^{-1}\tilde{g}(st),\tilde{\alpha }(st))$ smoothly converges to an expander $(g^{\ast }(t),{\alpha }^{\ast }(t))$ on ${\mathbb{R}}^2$ as $s\to \mathrm{\infty }$ where ${\alpha }^{\ast }(t)$ is nonzero and constant in spacetime and $g^{\ast }(t)$ is flat and expanding in the ${\alpha }^{\ast }$ direction. If $[\alpha ]$ is irrational, then there is a sequence $s_i\to \mathrm{\infty }$ such that $(M,s_i^{-1}g(s_{i}t))$ Gromov–Hausdorff converges to a point. Moreover, on the universal cover, $(s_i^{-1}\tilde{g}(s_{i}t),\tilde{\alpha }(s_{i}t))$ smoothly converges to $(g^{\ast }(t),{\alpha }^{\ast }(t))$.