High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres

We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\ge 2$ and let $\mathrm{\Sigma }$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_{\mathrm{\ell }}\to +\mathrm{\infty }$ such that, for every $\mathrm{\ell }$, there exists a complex manifold of complex dimension $2d_{\mathrm{\ell }}+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $\mathrm{\Sigma }$ into ${\mathbb{S}}^{2n}$ of degree $d_{\mathrm{\ell }}$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.