The Lawson surfaces are determined by their symmetries and topology

We prove that a closed embedded minimal surface in the round three-sphere which satisfies the symmetries of a Lawson surface and has the same genus is congruent to the Lawson surface.Comment: 15 pages, no figures, with minor improvement

Disc stackings and their Morse index

We construct free boundary minimal disc stackings, with any number of strata, in the three-dimensional Euclidean unit ball, and prove uniform, linear lower and upper bounds on the Morse index of all such surfaces. Among other things, our work implies for any positive integer $k$ the existence of $k$-tuples of distinct, pairwise non-congruent, embedded free boundary minimal surfaces all having the same topological type. In addition, since we prove that the equivariant Morse index of any such free boundary minimal stacking, with respect to its maximal symmetry group, is bounded from below by (the integer part of) half the number of layers and from above strictly by twice the same number, it follows that any possible realization of such surfaces via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters. This also shows that it is only for $N=2$ and $N=3$ layers that free boundary minimal disc stackings are achievable by means of one-dimensional mountain pass schemes.Comment: 55 pages, 8 figure

Second-order mass estimates for static vacuum metrics with small Bartnik data

Given on the 2-sphere Bartnik data (prescribed metric and mean curvature) that is a small perturbation of the corresponding data for the standard unit sphere in Euclidean space, we estimate to second order, in the size of the perturbation, the mass of the asymptotically flat static vacuum extension (unique up to diffeomorphism) which is a small perturbation of the flat metric on the exterior of the unit ball in Euclidean space and induces the prescribed data on the boundary sphere. As an application we obtain a new upper bound on the Bartnik mass of small metric spheres to fifth order in the radius

The Lawson surfaces are determined by their symmetries and topology

We prove that a closed embedded minimal surface in the round three-sphere which satisfies the symmetries of a Lawson surface and has the same genus is congruent to the Lawson surface.ISSN:0075-4102ISSN:1435-534

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

The topology and symmetry group of a free boundary minimal surface in the three-dimensional Euclidean unit ball do not determine the surface uniquely. We provide pairs of non-isometric free boundary minimal surfaces having any sufficiently large genus $g$, three boundary components and antiprismatic symmetry group of order $4(g+1)$.Comment: final preprint version, to appear in Memoirs of the American Mathematical Societ