Isothermic constrained Willmore tori in 3-space. (English) Zbl 1475.53068
Willmore surfaces in the 3-sphere \(S^3\) are critical points of the Willmore functional:
\[\mathcal W(f)=\int_M(H^2+1)dA,\]
where \(f:M\to S^3\) is an immersion from an oriented surface \(M\) into \(S^3\), \(H\) is the mean curvature and \(dA\) is the induced area form of \(f\). Critical points of the Willmore functional restricted to a given conformal class are called constrained Willmore surfaces, and surfaces admit conformal curvature line parametrizations away from their umbilical points are called isothermic. In this paper, the authors prove that isothermic constrained Willmore tori in the conformal 3-sphere with Willmore energy below \(8\pi\) are constant mean curvature surfaces in the round 3-sphere.
Reviewer: Atsushi Fujioka (Osaka)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A31 | Differential geometry of submanifolds of Möbius space |
Keywords:
constrained Willmore tori; spectral curve; quaternionic Plücker estimate; isothermic surfaceReferences:
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