Umbilicity and characterization of Pansu spheres in the Heisenberg group

Abstract

Abstract For every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>2</m:mn></m:mrow></m:math> {n\geq 2} we define a notion of umbilicity for hypersurfaces in the Heisenberg group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>H</m:mi><m:mi>n</m:mi></m:msub></m:math> {H_{n}} . We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p -mean (or horizontal-mean) curvature in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>H</m:mi><m:mi>n</m:mi></m:msub></m:math> {H_{n}} up to Heisenberg translations.