The support function defined on the unit sphere can be locally represented as a support function defined in the entire space. This can be achieved by using a mapping \begin{align} (x_1,\cdots,x_n)&\rightarrow\frac{(x_1,\cdots,x_n,-1)}{(1+|x|²)^{\frac{1}{2}}},\\ \mathbb{R}^n&\rightarrow \mathbb{S}^n_- \end{align} to obtain a local coordinate system for $\mathbb{S}^n_-$. In this coordinate system the metric $\hat{g}$ is given by \begin{align} \hat{g}^{ij}=(1+|x|^2)^{-1}\left(\delta_{ij}-\frac{x_ix_j}{1+|x|^2}\right). \end{align} From this, we can take the standard basis $e_1,\cdots,e_n$ with respect to this metric. Let $H$ be the support function of the convex hypersurface $ M $, and set \begin{align} u(x_1,\cdots,x_n)=(1+|x|^2)^{\frac{1}{2}}H\left(\frac{x_1,\cdots,x_n,-1}{(1+|x|^2)^{\frac{1}{2}}}\right). \end{align}
By the degree one homogeneity of $H$, $u$ is just the restriction of $H$ to $x_{n+1}=-1$. We use $D$ to represent the Euclidean space $\mathbb{R}^n$ gradient and $\hat{\nabla}$ to represent the gradient with respect to the unit basis on the unit sphere $\mathbb{S}^n$.
I see that in the paper, there are relationships for second-order derivatives: \begin{align} \hat{\nabla}_1\hat{\nabla}_1H=\frac{\partial^2}{\partial x_1\partial x_1}u\frac{(1+|x|^2)^{\frac{3}{2}}}{1+|x|^2-x_1^2}. \end{align}
The above information is taken from the papers： "Urbas J I E. An expansion of convex hypersurfaces[J]. Journal of Differential Geometry, 1991, 33(1): 91-125." and "Chou K S, Wang X J. A logarithmic Gauss curvature flow and the Minkowski problem[J]. Annales de l'Institut Henri Poincaré C, 2000, 17(6): 733-751.".
I want to know the relationship (an equation) between these two gradients ($Du$ and $\hat{\nabla}H)$. This question has been puzzling me for a long time, and I would like to seek advice from everyone. Thank you very much for your help.