The Costa surface with a free-boundary hole: an embedded exterior free-boundary minimal surface with three ends

An exterior free-boundary minimal surface is a properly immersed minimal surface outside the unit ball whose compact boundary lies on the unit sphere and meets it orthogonally. By Mazet and Mendes, in $\mathbb{R}^3$ every such surface with one regular end is a catenoidal piece, and every stable one with parallel embedded regular ends is catenoidal. We construct connected embedded examples with three ends from the Costa surface $M$, placed so that its central saddle point is at the origin. For every sufficiently small $\varepsilon>0$ we perturb $\varepsilon^{-1}(M\setminus B_\varepsilon)$ into a properly embedded exterior free-boundary minimal surface of genus one with one boundary circle and three ends, two catenoidal and one planar. The construction is a small-hole perturbation. The only error of the initial surface is in the boundary condition. The symmetry group of the Costa surface removes all obstructions: at the hole it forbids the logarithmic and the tilting modes, and globally it kills the bounded Jacobi fields of the Costa surface, which is nondegenerate by Nayatani. In the conformal compactification the Jacobi operator is smooth also at the ends, so the linear analysis takes place on a compact torus with one small hole. The same construction applied to the Costa-Hoffman-Meeks surfaces gives such examples of every genus $k\geq1$, of index at least $2k+3$. We also report numerical experiments with two independent implementations of the Costa surface. They confirm the period condition, the symmetries, the boundary error, the index and nullity of the Jacobi operator and the absence of an equivariant kernel. They also compute the free-boundary correction near the hole, which gives pictures of the surfaces.