Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to \mathbb{R}$ be a function.
I want a formula for $\Delta_{S(t)}u(x)$ in terms of $\Delta_{S(0)}$ and $u(F^0_t(y))$ where $x=F^0_t(y)$:
$$\Delta_{S(t)}u(x) = ??$$
By nice I am hoping for something that does not involve huge sums explicitly. I'm sure this kind of issue is well-established but I can't find anything.
Note that the surface gradient is defined as $$\nabla_{S(\cdot)}v = \nabla v- \nabla v \cdot \nu(\cdot)\nu(\cdot)$$ where $\nu(\cdot)$ is a unit normal vector of $S(\cdot)$, and the Laplace-Beltrami is the surface divergence of the surface gradient as expected.