Flow through Lawson Surfaces
Nicholas SchmittMedia
Description
The Lawson surfaces $\xi_{p,q}$ are a family of minimal surfaces in $S^3$ of genus $pq$. There is a flow through minimal surface in which the two integers are replaced by real parameters.
Shown in the images is one leg of this flow, e.g. from the Lawson surface $\xi_{2,1}$ of genus 2 to $\xi_{2,2}$ of genus 4. An order 2 symmetry of the initial surface with six fixed points is “opened” along three cuts until it reaches an order 3 symmetry. The final surface is show before and after the missing piece is filled in.
Also a flow from the Clifford torus $\xi_{1,1}$ of genus 1 to the Lawson minimal surfaces $\xi_{1,2}$ of genus 2 can be seen. Note, that the flow breaks the topology. Moreover, flows from a Delaunay torus to a Lawson CMC surface of genus 2 are shown.
References
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Herbert B. Lawson.
Complete minimal surfaces in $S^3$.
The Annals of Mathematics, 1970.
URL: http://www.math.jhu.edu/~js/Math748/lawson.s3.pdf.