For a broad class of cones in R^3, complete connected self-expanders of any positive genus exist, providing mean curvature flows in which genus drops but does not reach zero.
Allard, On the first variation of a varifold , Ann
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Self-expanders of positive genus
For a broad class of cones in R^3, complete connected self-expanders of any positive genus exist, providing mean curvature flows in which genus drops but does not reach zero.