Wente Torus

In 1894, Henry Wente discovered the Wente torus, an immersed torus of constant (but not zero) mean curvature. Such constant mean curvature (CMC) surfaces are counterexamples to Hopf's conjecture that a sphere is the only closed surface with constant mean curvature that is compact [1]. The set of all symmetric tori is in one-to-one correspondence with the set of reduced fractions ; each corresponds to a symmetric Wente torus labeled as [2].


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[1] H. C. Wente, "Counterexample to a Conjecture of H. Hopf," Pacific Journal of Mathematics, 121(1), 1986 pp. 193–243. projecteuclid.org/euclid.pjm/1102702809.
[2] Marija Ćirić, "Notes on Constant Mean Curvature Surfaces and Their Graphical Presentation," Filomat 23(2), 2009 pp. 97–107. www.pmf.ni.ac.rs/pmf/publikacije/filomat/2009/23-2-2009/Paper10.pdf.
[3] M. Heil. CMC: Pictures of Constant Mean Curvature Tori [Video]. (Jun 16, 2014) www.youtube.com/watch?v=7rnsdcS7qGU.
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