For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image of $\nabla$ is equal to $1$.

Now assume that $M$ is an open manifold(non compact without boundary). What type of results exsist for the image of $\nabla$. Is it surjective? Is the codimension of its image finite?