Disclaimer: The only course that I have seen in differential geometry is an introduction to differential geometry of manifolds and so I've never dealed with Laplace-Beltrami operator in the past.
I currently started to study a paper and at a point I came across with the following:
Let $\Gamma$ be a $C^2-$regular boundary of an open, bounded and connected subset of $\mathbb R^3$. Define the function: $\widetilde u: B(x_0,r) \cap \Gamma \to \mathbb R$ as: $\widetilde u=\frac{(\alpha +1)\theta}{16}{\vert x-x_0\vert}^2$
Then the Laplace-Beltrami operator of $\widetilde u$ is given by: $-\Delta \widetilde u= \frac{(\alpha +1)\theta}{16}(-4- H(x) \cdot (x-x_0))(*)$ where $H$ denotes the mean curvature vector of $\Gamma$.
After some research I found the following formula for calculating the Laplace-Beltrami operator:
$\Delta_{\Gamma} f=\Delta f-{\nabla}^2 f(\eta,\eta)+H_{\Gamma}\nabla f$ where $\Delta$ is the usual Euclidean Laplacian and ${\nabla}^2 $ denotes the hessian.
I would really appreciate if somebody could explain to me why the term ${\nabla}^2 \widetilde u (\eta,\eta)$ is missing from $(*)$.
Thanks in advance!