I asked the same question but get no answer in other place. Here is the following.

For a compact Riemannian surface $\Sigma$. For an initial embedded closed curve $\gamma_0$ in $\Sigma$, a family $\gamma_t$ $(0\leq t<T)$ is parametrized by \begin{equation} F : S^{1} \times[0, T) \rightarrow \Sigma, \end{equation} it is called a curve shortening flow, if \begin{equation} \frac{\partial}{\partial t} F(\theta, t)=-\kappa_{t}(F(\theta, t)) v_{t}(F(\theta, t)) \end{equation} P. Topping states on page 51 in

P. Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math. 503, 47-61, 1998

https://www.degruyter.com/view/j/crll.1998.1998.issue-503/crll.1998.099/crll.1998.099.xml

that \begin{equation} \frac{d A_{t}}{d t}=-\int_{\gamma_{t}} \kappa_{t}. ~~~(1) \end{equation} where $A_t$ is the area of the set bounded by the curve $\gamma_t$.

I know how to derive this for $\Sigma=\mathbb{R}^2$. How to prove (1) for $\Sigma$ being a surface? Thank you very much.

I asked the same question but get no answer in other place. Here is the following.

For a compact Riemannian surface $\Sigma$. For an initial embedded closed curve $\gamma_0$ in $\Sigma$, a family $\gamma_t$ $(0\leq t<T)$ is parametrized by \begin{equation} F : S^{1} \times[0, T) \rightarrow \Sigma, \end{equation} it is called a curve shortening flow, if \begin{equation} \frac{\partial}{\partial t} F(\theta, t)=-\kappa_{t}(F(\theta, t)) v_{t}(F(\theta, t)) \end{equation} P. Topping states on page 51 in [1] that \begin{equation} \frac{d A_{t}}{d t}=-\int_{\gamma_{t}} \kappa_{t}. ~~~(1) \end{equation} where $A_t$ is the area of the set bounded by the curve $\gamma_t$.

I know how to derive this for $\Sigma=\mathbb{R}^2$. How to prove (1) for $\Sigma$ being a surface? Thank you very much.

References

[1] P. Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math. 503, 47-61, 1998