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Biharmonic surfaces of constant mean curvature. (English) Zbl 1322.53060

A map \(\phi:(M^m, g)\to (N^n,h)\) between Riemannian manifolds is a biharmonic map if it is a critical point of the bienergy functional \(E_2(\phi)=\frac{1}{2}\int_M|\tau(\phi)|^2dv_g\), where \(\tau(\phi)\) is the tension field of the map \(\phi\). Equivalently, \(\phi\) is a biharmonic map if and only if its bitension field \[ \tau_2(\phi):=-\Delta \tau(\phi)-\text{trace}_g\, \text{R}^N (\text{d}\phi(\cdot), \tau(\phi)) \text{d}\phi(\cdot)=0 \] vanishes identically. The biharmonic stress energy tensor of the map \(\phi:(M^m, g)\to (N^n,h)\) is defined to be \[ S_2(X,Y)=[\frac{1}{2}|\tau(\phi)|^2+\langle \text{d}\phi, \nabla\tau(\phi) \rangle] g(X, Y) -\langle \text{d}\phi(X), \nabla_Y\tau(\phi) \rangle-\langle \text{d}\phi(Y), \nabla_X\tau(\phi) \rangle. \] An interesting link between the bitension field and the biharmonic stress energy tensor was given by G. Jiang in [Acta Math. Sin. 30, No. 2, 220–225 (1987; Zbl 0631.58007)] as: \(\operatorname{div} S_2=-\langle\text{d} \phi, \tau_2(\phi)\rangle\).
In the paper under review, the authors compute the rough Laplacian of the biharmonic stress energy tensor and use it to proved some rigidity results for biharmonic constant mean curvature surfaces.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.

Citations:

Zbl 0631.58007
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