## Bounds on the topology and index of minimal surfaces.(English)Zbl 1428.53018

Summary: We prove that for every non-negative 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.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q05 Minimal surfaces and optimization 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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