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A characterization of constant \(p\)-mean curvature surfaces in the Heisenberg group \(H_1\). (English) Zbl 1506.53070

The Heisenberg group \(H_1\) is \(\mathbb{R}^3\) with the multiplication \((x_1,y_1,z_1)\circ(x_2,y_2,z_2)= (x_1+ x_2, y_1+ y_2, z_1+z_2+ y_1x_2- x_1 y_2)\). The authors prove that the existence of a constant \(p\)-mean curvature surface (without singular points) is equivalent to the existence of a solution of a specific nonlinear second-order equation.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C22 Geodesics in global differential geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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