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We study immersed, connected, umbilic hypersurfaces in the Heisenberg group ( H_{n}) with n (ge ) 2. We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.

Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group