"It is known that, for a curvature continuous surface, if the curve $\gamma$ shrinks to the point $v_i$, the (line) integral converges to the mean curvature $\kappa(v_i)$ of the surface at the point $v_i$ times the normal vector $N_i$ at the same point " $$\lim_{\epsilon \to 0} \frac{1}{|\gamma_\epsilon|}\int_{v\in\gamma_\epsilon} (v-v_i)dl(v) = \kappa(v_i)N_i $$ Where $\gamma_\epsilon$ is a closed curve embedded in the surface which encircles the vertex $v_i$ and $|\gamma_\epsilon|$ is the length of the curve. Especially $|\gamma_\epsilon|\to0$. From Gabriel Taubin "A Signal Processing Approach To Fair Surface Design" .

It is states as a common fact from differential geometry. But it is not clear to me. How would one prove this statement?

Does it has something to do with $$\kappa(v_i) N_i = \lim_{r(A)\to 0} \frac{3\nabla(A)}{2A} $$ where $r(A)$ is the diameter of a small region around $v_i$. $\nabla$ is the gradient with respect $x,y,z$ the coordinates.

M.Do Carmo. "Differential Geometry of Curves and Surfaces" is cited but I could not find anything which helped me prove it.

He is using this to define a normal vector for polyhedral surfaces via the discrete Laplacian, which can be understood as an approximation of the above line integral. Also the second expression is used to approximate the Laplace-Beltrami operator for triangulated surfaces.