I assume that what you mean by "Fermi surface without the periodic potential" is something like the free-particle Fermi surface. If the periodic potential is weak and your original Fermi surface is not close to the (Brillouin Zone) BZ zone boundary, the Fermi surface may only change a little bit because the weak periodic potential only fold your whole energy spectrum back to the first BZ. The Fermi surface will change significantly when the periodic potential is not weak.