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To state my question, first I give some notation. Let $(M,g)$, $(N,h)$ be compact manifolds. A map $u:M \times [0,T) \to N$ is called a solution of the harmonic map heat flow if satisfies $$\frac{\partial}{\partial t}u=\Delta u,$$ on $M \times [0,T)$, where $\Delta u$ is the intrinsic Laplacian (or the map Laplacian, of the tension field) given in local coordinates by $$\Delta u^{k}=g^{\alpha \beta} \left( \frac{\partial^{2}u^{k}}{\partial x_{\alpha}\partial x_{\beta}}-\frac{\partial u^{k}}{\partial x_{\gamma}}\Gamma_{\alpha \beta}^{\gamma}(x)+\frac{\partial u^{i}}{\partial x_{\alpha}}\frac{\partial u^{j}}{\partial x_{\beta}} \Gamma_{ij}^{k}(u) \right)=\Delta_{M} u^{k}+g^{\alpha \beta}\frac{\partial u^{i}}{\partial x_{\alpha}}\frac{\partial u^{j}}{\partial x_{\beta}} \Gamma_{ij}^{k}(u) ,$$ where $\Gamma_{\alpha \beta}^{\gamma}$ are the Christoffel symbols of $M$ and $\Gamma_{ij}^{k}$ are the Christoffel symbols of $N$, and $\Delta_{M}$ is the Laplace-Beltrami operator in $(M,g)$. Recall also that given tensors $A$ and $B$, $A*B$ is defined to be any linear combination of contractions of $A \otimes B$.

In the paper "The Formation of Singularities in the Harmonic Map Heat Flow" of Grayson & Hamilton, they prove a bound for the higher derivatives of a solution of the harmonic map heat flow. For this, in the proof the autors says that he have the following formula \begin{equation} \tag{1} \frac{\partial}{\partial t} DF=\Delta DF+R_{M}*DF+R_{N}(F)*DF^{3}, \end{equation} Here, $DF$ should be the usual differential of $F:M\times [0,T) \to N$, where $F$ satisfies the harmonic map heat flow. Also, the say that differentiating (1), one obtain \begin{equation}\tag{2}\frac{\partial}{\partial t} D^{2}F=\Delta D^{2}F+R_{M}*D^{2}F+R_{N}(F)*DF^{2}*D^{2}F+DR_{M}*DF+DR_{N}(F)*DF^{4}.\end{equation} Here comes my question. How can obtain this two equations? I suppose that (1) comes from something like the following $$\frac{\partial}{\partial t}DF=\frac{\partial}{\partial t} \frac{\partial F^{k}}{\partial x_{\delta}}=\frac{\partial}{\partial x_{\delta}} \frac{\partial F^{k}}{\partial t}=\frac{\partial}{\partial x_{\delta}}\Delta F^{k}.$$ I tried to expand the RHS but I do not be able to obtain (1). Also, to obtain (2) from (1), I have a problem with the notation. I think that $D^{2}F$ should mean $\nabla DF$, and so, to obtain (2), one should take the covariant derivative to (1) to obtain (2).

Thanks!

EDIT: I know that some evolutions equations exists for others flow, e.g. mean curvature flow (from the classical paper of Hiusken and also see "Lecture Notes on Mean Curvarute Flow" by Mantegazza) and for Ricci flow (see lema 13.1 of "Three-manifolds with positive Ricci curvature" by Hamilton), so I suppose that this calculations are "well known".

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1 Answer 1

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Some time ago I found a reference which explains these equations. In the book The Ricci Flow in Riemannian Geometry written by Ben Andrews and Christopher Hopper you can find a derivation of (1) and (2):

  • Equation (1) is the evolution equation for $f_{*}$ in page 41 of the book (63 of the pdf) but written in Hamilton's notation.
  • Equation (2) is the evolution equation for $\nabla f_{*}$ in page 44 of the book (66 of the pdf).

Note that Grayson & Hamilton use $DF$ and $D^{2}F$ for what Andrews and Hopper write as $f_{*}$ and $\nabla f_{*}$, respectively.

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