Let $F_t:\Sigma^n\to\mathbb{R}^{n+1}$ be a smooth family of immersion of a hypersurface that satisfies the inverse curvature flow: \begin{align} \tag1 \frac{\partial F_t}{\partial t}=\frac{1}{f}\nu. \end{align} Now, suppose each $\Sigma_t:=F_t(\Sigma)$ is a star-shaped hypersurface in $\mathbb{R}^{n+1}$; that is, there exists a smooth positive function $r$ such that \begin{align} \Sigma_t=\left\{(r(\theta,t),\theta):\theta\in\mathbb{S}^n\right\} \end{align} Let $(\theta^i)$ be a local coordinate system of $\mathbb{S}^n$, and denote $\sigma_{ij}$ the components of the round metric on $\mathbb{S}^n$ with respect to $(\theta^i)$. Let $\nabla$ be the Levi-Civita connection on $\mathbb{S}^n$ with respect to the round metric, and let $|\cdot|$ be the norm induced by the round metric. Denote \begin{align} \rho:=\sqrt{1+|\nabla\log r|^2} \end{align} If we write the flat metric on $\mathbb{R}^{n+1}$ by $\delta=dr\otimes dr+r^2g_{\mathbb{S}^n}$, then the unit normal vector on $\Sigma_t$ can be given by \begin{align} \nu=\frac{1}{\rho}\left(\partial_r-(\nabla^i\log r)\partial_i\right) \end{align}

Now, the inverse curvature flow can be reduced to the following equation: \begin{equation} \tag2 \frac{\partial r}{\partial t}=\frac{\rho}{f} \end{equation} Indeed, we have the following computation: \begin{align} \frac{1}{f}&=\left\langle\frac{\partial F_t}{\partial t},\nu\right\rangle \\ &=\left\langle\frac{\partial r}{\partial t}\partial_r,\frac{1}{\rho}\left(\partial_r-(\nabla^i\log r)\partial_i\right)\right\rangle \\ &=\frac{1}{\rho}\frac{\partial r}{\partial t}\left[\underbrace{\langle\partial_r,\partial_r\rangle}_{=1}-\nabla^i\log r\underbrace{\langle\partial_r,\partial_i\rangle}_{=0}\right] \\ &=\frac{1}{\rho}\frac{\partial r}{\partial t} \end{align}

Here comes my question: What is the error in the following computation: \begin{align} \frac{\partial r}{\partial t}&=\left\langle\frac{\partial r}{\partial t}\partial_r,\partial_r\right\rangle \\ &=\left\langle\frac{\partial F_t}{\partial t},\partial_r\right\rangle \\ &=\left\langle\frac{1}{f}\nu,\partial_r\right\rangle \\ &=\frac{1}{f}\left\langle\frac{1}{\rho}\left(\partial_r-(\nabla^i\log r)\partial_i\right),\partial_r\right\rangle \\ &=\frac{1}{\rho f} \end{align} which obviously does not get the same answer as the computation above?

For reference, I was reading "Flow of Nonconvex Hypersurfaces Into Spheres" by Claus Gerhardt. The equation (2) that I wish to obtain is the equation (1.8) in his paper. In this post, I have reformulated the whole problem in my own language and notations, while also omitted some detail which I think is not relevant to my problem here. I hope my writing is correct and I apologise if this brings any inconvenience to you all.

Thanks in advance for any comment and answer.


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