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Willmore's conjecture for immersed tori

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Let Φ⃗:T2→Rm\vec{\Phi}:T^2\to\mathbb{R}^m be an immersion, and let H⃗Φ⃗\vec{H}_{\vec{\Phi}} and dvol⁡Φ⃗∗gRmd\operatorname{vol}_{\vec{\Phi}^*g_{\mathbb{R}^m}} denote its mean curvature vector and induced area element. Willmore's conjecture. One has

∫T2∣H⃗Φ⃗∣2 dvol⁡Φ⃗∗gRm≥2π2.\int_{T^2}|\vec{H}_{\vec{\Phi}}|^2\,d\operatorname{vol}_{\vec{\Phi}^*g_{\mathbb{R}^m}}\geq 2\pi^2.

Equality should hold only when Φ⃗(T2)\vec{\Phi}(T^2) is a Möbius transform of the stereographic projection into R3\mathbb{R}^3 of the Clifford torus

Tcliff2:={12(eiθ,eiϕ)∈C2:(θ,ϕ)∈R2}⊂R3.T^2_{\mathrm{cliff}}:=\left\{\frac{1}{\sqrt{2}}(e^{i\theta},e^{i\phi})\in\mathbb{C}^2:(\theta,\phi)\in\mathbb{R}^2\right\}\subset\mathbb{R}^3.

The conjecture is the central lower-bound and rigidity assertion for the Willmore energy of immersed tori. The supplied source does not state a resolution status.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Willmore's conjecture for immersed tori

    Let F:N→RnF:N\to\mathbb{R}^n be an immersion of a closed surface into Euclidean space, with induced metric, area form darea\mathop{\mathrm{darea}}, mean curvature HH, and Willmore integral

    W(F):=∫NH2 darea.\mathcal{W}(F):=\int_N H^2\,\mathop{\mathrm{darea}}.

    Let T2T^2 denote the two-dimensional torus. Willmore's conjecture. For any immersion F:T2→RnF:T^2\to\mathbb{R}^n with n≥3n\geq 3, the inequality

    W(F)≥2π2\mathcal{W}(F)\geq 2\pi^2

    holds. The conjecture extends the sharp lower bound for immersed spheres and predicts the minimum Willmore energy among immersed tori. The source states that it remains open at the time of writing, while proving it under the condition that the LpL^p-norm of the Gaussian curvature is sufficiently small.

    source: Bernd Ammann, “The Willmore Conjecture for immersed tori with small curvature integral”, arXiv:math/9906065 (1999).

References

Primary source

Tristan Rivière, “Conformally Invariant Variational Problems”, arXiv:1206.2116 (2012).

differential-geometryimmersed-surfacesminimal-surfaceswillmore-energywillmore-surfaces
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