Solvability of $u_t=\nabla_{u_\theta} u_\theta -\frac{1}{2}\nabla R +\frac{1}{2\tau}u_\theta + 2 Ric(u_\theta,\cdot)$

Question

I ask a question in math.stackexchange, but nobody answer it. So, I ask here.

In Cao and Zhu's paper about Ricci flow, the $\mathcal{L}$-geodesic is defined as picture below.In fact, $\mathrm{Ric}(X,\cdot)$ should be $g^{-1}\mathrm{Ric}(X,\cdot)$.

I want to study the heat flow of it, similar to the heat flow of ordinary geodesic. Let $u(t,\tau)$ be a family of curves, $t\ge0$ and for any $t_0\ge 0$, $u(t_0,\cdot):S^1\rightarrow M$ is a curve on $M$. $(M,g)$ is a compact Riemannian manifold. $$ u_t=\nabla_{u_\tau} u_\tau -\frac{1}{2}\nabla R +\frac{1}{2\tau}u_\tau + 2 \mathrm{Ric}(u_\tau,\cdot) $$ In fact, I want to construct some convexity energy to prove the existence of solution. But fail. And fail for simple situation $$ u_t=\nabla_{u_\tau} u_\tau +\nabla R $$ I feel the method of energy is not suitable for it , because energy function is hard to construct . What method is suitable for this question ?

Answer 1

1. In Ottarsson's paper that you cited the curves are by definition closed, that is, they are images of $\mathbb{S}^1$. In Cao and Zhu, the $\mathcal{L}$-geodesics must be "open" in the sense that they are images of $[\tau_1,\tau_2]$. So if you want to write down any equation you absolutely have to write down boundary conditions. I assume you want something like Dirichlet where the end points are fixed.

2. The heat flow given in Ottarsson's paper is the gradient flow relative to the energy functional for immersions of $\mathbb{S}^1$ to $M$. If you want to define an analogous object you should have the gradient flow relative to the $\mathcal{L}$-length, and it seems to me that your definition is missing a factor of $\sqrt{\tau}$ on the right hand side of the flow equation.

3. In terms of the $\mathcal{L}$-length, you are using $\tau$ as the parameter for your curve, so all your $\theta$s should really be $\tau$s.

4. In terms of the basic energy: you are working with the gradient flow of an energy functional. Just use that as the energy.