Bounds on the topology and index of minimal surfaces

17 Sep 2019 · Meeks William H. III, Perez Joaquin, Ros Antonio ·

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.