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Higher genus minimal surfaces in $$S^3$$ and stable bundles. (English) Zbl 1288.53007
This is one of four papers (so far) of the author investigating embedded genus-two minimal surfaces in $$\mathbb{S}^3$$. The main tools are the Dorfmeister-Pedit-Wu representation and Hitchin’s gauge-theoretic approach to minimal surfaces in $$\mathbb{S}^3$$. Particular attention is given to surfaces that have the same symmetry of Lawson’s genus-two example. In that case the results yield (up to some period and closing conditions) explicit formulas. Many of the ideas continue to hold for surfaces of any genus and any constant mean curvature in $$\mathbb{S}^3$$ or $$\mathbb{R}^3$$. These investigations open up an entirely new and promising direction in the general area of “soliton geometry”.

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
minimal surfaces
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