We present new examples of complete embedded self-similar surfaces under mean curvature flow by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of nonrotationally symmetric self-shrinkers in . Although our initial approximating surfaces are asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.
"Construction of complete embedded self-similar surfaces under mean curvature flow, part III." Duke Math. J. 163 (11) 2023 - 2056, 15 August 2014. https://doi.org/10.1215/00127094-2795108