On a Bers theorem for SL (3, C)

For S a closed surface of genus at least 2, let Hit₃ (S) be the Hitchin component of representations to SL (3, R), equipped with the Labourie-Loftin complex structure. We construct a mapping class group equivariant holomorphic map from a large open subset of Hit₃ (S) Hit₃ (S) to the SL (3, C) -character variety that restricts to the identity on the diagonal and to Bers' simultaneous uniformization on T (S) T (S). The open subset contains Hit₃ (S) T (S) and T (S) Hit₃ (S), and the image includes the holonomies of SL (3, C) -opers. The map is realized by associating pairs of Hitchin representations to immersions into C³ that we call complex affine spheres, which are equivalent to certain conformal harmonic maps into SL (3, C) /SO (3, C) and to new objects called bi-Higgs bundles. Complex affine spheres are obtained by solving a second-order complex elliptic PDE that resembles both the Beltrami and Tzitz\'eica equations. To study this equation we establish analytic results that should be of independent interest.