Examples of Stable Embedded Minimal Spheres without Area Bounds

This answers affirmatively a question posed in [CM04]. In [CM99], T. H. Colding and W. P. Minicozzi II proved an analogous theorem, stating that there exists an open set of metrics on M such that there are embedded minimal tori of arbitrarily large area. This result was extended to surfaces of positive genus in [Dea03] by B. Dean. The genus zero case has remained an open problem since then, largely due to the fact that the fundamental group of a sphere is trivial. The approaches to building the positive genus surfaces followed a simple structure. One considers a closed three-dimensional submanifold of the unit three-ball which is thought of as a subset of the larger manifold. A sequence of surfaces is then constructed such that the area, allowing for certain variance of the surface, seems to approach infinity. Given any element of this sequence, a result by R. Schoen and S. T. Yau (see [SY79]) is used to show that there is a stable minimal surface which closely approximates this element. The final step is to show that the area becomes unbounded. The result of Schoen and Yau mentioned above states that given a closed non-simply connected embedded surface with for which the inclusion map induces an injection of the fundamental group, one can always find a closed embedded stable minimal surface of the same genus whose fundamental group has the same image under the induced map from inclusion. Replacing this is a result by W. Meeks III, L. Simon, and S. T. Yau (see [MSY82]) in which one can find a set of closed stable embedded minimal surfaces which is obtained from the original by pinching off parts of the surface and varying each isotopically. This is discussed in detail in section (4). It should also be noted that J. Hass, P. Norbury, and J. H. Rubinstein constructed embedded minimal spheres of unbounded morse index in [HNR03] by methods which are significantly different than those presented here. Also, in contrast of the main theorem, H. Choi and A. Wang proved in [CW83] that there is an open set of metrics, namely those with positive ricci curvature, for which there is a uniform area bound on compact embedded minimal surfaces depending only on the genus and the ambient metric. The author also suspects that it may be possible to provide a bound (depending on the metric) for the genus of any embedded minimal surface in the unit ball.