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  $\Delta$ is equal to $1$.

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