Let L:K^2→S^3 be Lawson's minimal Klein bottle, let sigma:S^3 setminusp→R^3 be a stereographic projection for a point p not on L(K^2), and set f_L=sigma circ L. Kusner's conjecture asserts that W(f)≥W(f_L) for every smooth immersion f:K^2→R^3; equivalently, the stereographic image of Lawson's minimal Klein bottle minimizes the Willmore energy among immersed Klein bottles in R^3.

Kusner’s conjecture; Hirsch–Mäder–Baumdicker conjecture

Let $L:\mathbb{K}^2\to\mathbb{S}^3$ be Lawson's minimal Klein bottle, let $\sigma:\mathbb{S}^3\setminus\{p\}\to\mathbb{R}^3$ be a stereographic projection for a point $p$ not on $L(\mathbb{K}^2)$ , and set $f_L=\sigma\circ L$ . Kusner's conjecture asserts that $\mathcal{W}(f)\geq\mathcal{W}(f_L)$ for every smooth immersion $f:\mathbb{K}^2\to\mathbb{R}^3$ ; equivalently, the stereographic image of Lawson's minimal Klein bottle minimizes the Willmore energy among immersed Klein bottles in $\mathbb{R}^3$ .

References

Primary source

arXiv

Additional references

Willmore Energy Estimates for Klein Bottles — arXiv — Ruijie Ni, Peng Wang

differential-geometry willmore-energy

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new preprint proves the predicted lower bound over part of the parameter range and settles the flat Euclidean case, but the full three-dimensional conjecture remains open.

Kusner conjectured that the stereographic image of Lawson’s minimal Klein bottle minimizes Willmore energy among immersed Klein bottles in $\mathbb{R}^{3}$ . The related Hirsch–Mäder–Baumdicker prediction concerns the flat Euclidean minimizer in higher dimensions.

Known results

Earlier work establishes $\mathcal{W}(f)\ge 8\pi$ for immersed Klein bottles in $\mathbb{R}^{3}$ (2016).
For $n\ge 4$ , the infimum is attained by a smooth embedded Klein bottle and is below $8\pi$ (2016).
The stereographic projection of the bipolar Lawson surface uniquely minimizes in $\mathbb{R}^{n}$ for $n\ge 4$ , with value approximately $6.682\pi$ (2016).

September 2026 partial bound

Ruijie Ni and Peng Wang’s preprint claims the Kusner-type lower bound for $0.350\lesssim b\lesssim 0.755$ and confirms the flat-Klein-bottle prediction in Euclidean space. This advances the problem but does not establish Kusner’s conjecture outside that interval.

Current status (as of September 2026): The flat Euclidean case and the Kusner-type bound on $0.350\lesssim b\lesssim 0.755$ are claimed in a new preprint, while the full Kusner conjecture remains open.

Sources

No solutions have been posted yet.