Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group \(\mathbb H^n\)
For every positive integer \(n\) let \(\mathfrak{h}_{n}\) denote the Lie algebra of dimension \(2n+1\) with basis \(\mathfrak{B} = \{X_{1},Y_{1},\dots,X_{n},Y_{n},T \}\) and nonzero bracket relations \([X_{k},Y_{k}] = - [Y_{k},X_{k}]= -2T\). The \(2n+1\)-dimensional Heisenberg group \(H_{n}\) is the simply connected Lie group with Lie algebra \(\mathfrak{h}_{n}\) equipped with the left invariant Riemannian metric for which the basis \(\mathfrak{B}\) is orthonormal. The horizontal distribution in \(H_{n}\) is the left invariant distribution spanned by \(\{X_{1},Y_{1},\dots,X_{n},Y_{n} \}\). If \(S\) is a \(C^{1}\) hypersurface in \(H_{n}\), then the singular set \(S_{0}\) consists of those points \(p\) where the tangent space \(T_{p}S\) is horizontal. The singular set \(S_{0}\) is closed in \(S\) with nonempty interior. If \(U\) is a unit normal vector field on \(S\), then for \(p\in S - S_{0}\) one defines the mean curvature \(H(p)\) by \(-2nH(p) = \operatorname{div}U_{H}/| U_{H}| \), where \(U_{H}\) denotes the horizontal component of \(U\). Let \(O\) be an open subset of \(H_{n}\) with \(C^{1}\) boundary \(S\). Define \(P(O)\), the perimeter of \(O\), to be the Riemannian integral over \(S\) of \(| U_{H}| \). If \(O\) has finite perimeter, then \(O\) is said to be stationary if \(P'(0) = 0\) for all volume preserving variations \(O_{t}\), where \(P(t) = P(O_{t})\). The authors show that if \(O\) is stationary, then \(S\) has constant mean curvature. A hypersurface \(S\) is said to be rotationally symmetric if \(S\) is left invariant by the isometries of \(H_{n}\) that leave invariant the center \(Z = \exp(T)\). The hypothesis of rotational symmetry allows one to formulate the equations for constant mean curvature as a family of ordinary differential equations. The authors compute the mean curvature of a rotationally invariant hypersurface and several examples. They classify the rotationally invariant hypersurfaces of constant mean curvature into five basic types.
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