[Paper Review] Umbilicity and characterization of Pansu spheres in the Heisenberg group

Jih-Hsin Cheng, Hung-Lin Chiu+2 more|arXiv (Cornell University)|Jun 10, 2014

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper introduces a notion of umbilicity for hypersurfaces in the Heisenberg group $H_n$ ($n \geq 2$) based on the behavior of the horizontal shape operator and proves that Pansu spheres are the only compact, connected, $C^2$ hypersurfaces with positive constant $p$-mean curvature that are umbilic everywhere. The key result establishes a characterization of Pansu spheres via umbilicity, extending classical Alexandrov-type theorems to the sub-Riemannian setting using geometric analysis and CR geometry.

ABSTRACT

For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or horizontal)-mean curvature in $H_{n}$ up to Heisenberg translations.

Motivation & Objective

• To define a geometric notion of umbilicity for hypersurfaces in the Heisenberg group $H_n$ ($n \geq 2$) that generalizes the classical concept from Euclidean space.
• To classify umbilic hypersurfaces in $H_n$ under natural regularity and curvature conditions.
• To prove that Pansu spheres are the only compact, connected, $C^2$ hypersurfaces with positive constant $p$-mean curvature that are umbilic everywhere, up to Heisenberg translations.
• To extend the Alexandrov-type theorem to the sub-Riemannian setting by linking umbilicity and constant $p$-mean curvature.

Proposed method

• Define the horizontal normal $e_{2n} = Je_n$ and the vector field $X_n = \nabla_{e_n}e_{2n} + le_n$ on the regular part of the hypersurface, where $l$ is a scalar function related to curvature.
• Introduce the shape operator $-\nabla e_{2n} + \alpha J'$ acting on $\xi'$, the maximal $J$-invariant subspace of $\xi \cap T\Sigma$, and define umbilicity via its eigenvalues being equal.
• Use the invariance of $\xi'$ under the shape operator and the equality of eigenvalues $\lambda_1 = \cdots = \lambda_{2n-1}$ to define umbilic points.
• Apply Hopf's index theorem to show that a closed, compact hypersurface without singular points leads to a contradiction, implying the existence of singular points.
• Analyze the structure of the singular set $S_\Sigma$ and show it consists of isolated points, leading to the conclusion that $\Sigma \setminus S_\Sigma$ is connected and identified with a level set $\Sigma(K)$.
• Use the behavior of the function $\alpha$ near singular points to show that the only possible compact level set is the $\alpha$-axis, which corresponds to the Pansu sphere.

Experimental results

Research questions

• RQ1 Can a notion of umbilicity be defined for hypersurfaces in the Heisenberg group $H_n$ ($n \geq 2$) that generalizes the classical Euclidean concept?
• RQ2 Are Pansu spheres the only compact, connected, $C^2$ hypersurfaces in $H_n$ with positive constant $p$-mean curvature that are umbilic everywhere?
• RQ3 Does the existence of a singular point on a compact umbilic hypersurface in $H_n$ force the hypersurface to be congruent to a Pansu sphere?
• RQ4 Can the characterization of spheres via umbilicity in Euclidean space be extended to the sub-Riemannian setting of the Heisenberg group?

Key findings

• The paper defines a new notion of umbilicity in $H_n$ based on the invariance and equal eigenvalues of the shape operator $-\nabla e_{2n} + \alpha J'$ on $\xi'$.
• It proves that any compact, connected, $C^2$ hypersurface in $H_n$ with positive constant $p$-mean curvature and all regular points umbilic must contain at least one singular point.
• The existence of a singular point forces the function $\alpha$ to be bounded, which restricts the possible level sets to the $\alpha$-axis, implying the hypersurface is congruent to a Pansu sphere.
• The Pansu sphere is the only compact, connected, $C^2$ hypersurface in $H_n$ with positive constant $p$-mean curvature that is umbilic everywhere, up to Heisenberg translations.
• The paper shows that level sets of certain Sobolev extremal functions (e.g., for $p=2$ and $p=1$) are also umbilic, with $l=3k$ only when $\lambda=0$, i.e., exactly when the level set is a Pansu sphere.
• For $p=1$, the level sets of the solution to the $p$-mean curvature equation are umbilic and satisfy $l=2k$, which corresponds to the Pansu sphere structure.