I have an intuitive understanding of the gradient, divergence, curl, and Laplacian operators in multivariable calculus, but not of the vector Laplacian. Is there a visualizable intuition for the meaning of the Laplacian of a vector field on a Riemannian manifold?
And a closely related question: is there an intuitive explanation for what's "special" about solutions to the vector Laplace equation?
I found an answer. The property discussed at https://math.stackexchange.com/a/50285/268333 generalizes easily to the vector case for vector fields on $\mathbb{R}^d$:
$$ \nabla^2\, {\bf F}({\bf x}_0) = 2d \lim_{r \to 0^+} \frac{\langle {\bf F} \rangle_r - {\bf F}({\bf x}_0)}{r^2} , $$
where $\langle {\bf F} \rangle_r$ denotes the average value of the vector field over the small sphere of radius $r$ centered at ${\bf x}_0$. This means that the vector Laplacian of a vector field at a point is proportional to the leading-order amount by which the average value of the vector field near the point differs from its value at the point.