Minimal surfaces would provide a lot of examples.
Construction of minimal surfaces often reduces to finding a meromorphic and a holomorphic functions (Weierstrass representation). Often we have a general idea about these functions up to some parameters, which are determined by "closing the periods", meaning that some path integrals should vanish.
In many cases the periods are easily closed numerically, leading to beautiful pictures, and the surface is naturally named after the discoverer (they deserve). But existence of period-closing parameters could be very hard prove, sometimes never done. The Horgan (non)surface is a famous example that computer closes periods which is later proved impossible. Then there is the embeddedness to prove, which could be even harder.