We investigate the structure of the potential energy surfaces of the s
uperheavy nuclei Fm-258(158)100, (264)(156)Hs(108), (278)(166)112, (29
8)(184)114, and (292)(172)120 within the framework of self-consistent
nuclear models, i.e., the Skyrme-Hartree-Fock approach and the relativ
istic mean-field model. We compare results obtained with one represent
ative parametrization of each model which is successful in describing
superheavy nuclei. We find systematic changes as compared to the poten
tial energy surfaces of heavy nuclei in the uranium region: there is n
o sufficiently stable fission isomer any more, the importance of triax
ial configurations to lower the first barrier fades away, and asymmetr
ic fission paths compete down to rather small deformation. Comparing t
he two models, it turns out that the relativistic mean-field model giv
es generally smaller fission barriers. [S0556-2813(98)05510-1].