[Paper Review] Rigidity theorems of $λ$-hypersurfaces

Qing-Ming Cheng, Shiho Ogata+1 more|arXiv (Cornell University)|Mar 17, 2014

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper establishes rigidity theorems for complete $\lambda$-hypersurfaces in $\mathbb{R}^{n+1}$ with polynomial area growth by employing a generalized maximum principle for the $\mathcal{L}$-operator. It proves that such hypersurfaces are isometric to standard models—spheres, Euclidean spaces, or cylinders—unless a specific inequality involving the second fundamental form and mean curvature fails, in which case the hypersurface must be one of the standard types.

ABSTRACT

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with polynomial area growth. By making use of the generalized maximum principle for $\mathcal L$ of $λ$-hypersurfaces, we prove a rigidity theorem of complete $λ$-hypersurfaces.

Motivation & Objective

• To classify complete $\lambda$-hypersurfaces in $\mathbb{R}^{n+1}$ with polynomial area growth using geometric and analytic techniques.
• To extend rigidity results from self-shrinkers ($\lambda = 0$) to general $\lambda$-hypersurfaces.
• To establish a gap theorem and a rigidity theorem under the $\mathcal{L}$-operator generalized maximum principle.
• To characterize the structure of $\lambda$-hypersurfaces when the second fundamental form and mean curvature satisfy a critical inequality.
• To prove that under bounded curvature and polynomial area growth, $\lambda$-hypersurfaces must be isometric to standard models such as spheres, Euclidean spaces, or cylinders.

Proposed method

• Utilizes the $\mathcal{L}$-operator, defined as $\mathcal{L} = \Delta - \frac{1}{2}\langle X, \nabla \cdot \rangle$, on $\lambda$-hypersurfaces to analyze curvature and second fundamental form behavior.
• Applies a generalized maximum principle for $\mathcal{L}$-operators on complete $\lambda$-hypersurfaces with Ricci curvature bounded from below.
• Introduces the function $B = S - \frac{H^2}{n}$, where $S$ is the squared norm of the second fundamental form and $H$ is the mean curvature, to quantify deviation from umbilicity.
• Derives a key differential inequality involving $\mathcal{L}B$ and compares it to a threshold expression involving $\lambda$, $n$, and $H$.
• Employs integration of the identity $\frac{1}{2}\Delta|X|^2 = H\langle N,X\rangle + n$ over $M$ with respect to the Gaussian measure $e^{-|X|^2/2}d\mu$.
• Uses Stokes' theorem and $L^2$-integrability under polynomial area growth to derive pointwise identities and contradictions when curvature bounds are violated.

Experimental results

Research questions

• RQ1 Under what conditions is a complete $\lambda$-hypersurface with polynomial area growth isometric to a standard model such as a sphere, Euclidean space, or cylinder?
• RQ2 What is the role of the $\mathcal{L}$-operator generalized maximum principle in classifying $\lambda$-hypersurfaces?
• RQ3 How does the inequality $\left(\sqrt{S - \frac{H^2}{n}} + |\lambda|\frac{n-2}{2\sqrt{n(n-1)}}\right)^2 + \frac{1}{n}(H - \lambda)^2 \leq 1 + \frac{n\lambda^2}{4(n-1)}$ constrain the geometry of $\lambda$-hypersurfaces?
• RQ4 Can the rigidity of $\lambda$-hypersurfaces be established without assuming compactness, using only polynomial area growth and $L^2$-integrability?
• RQ5 What is the relationship between the second fundamental form, mean curvature, and the constant $\lambda$ in determining the isometry type of a $\lambda$-hypersurface?

Key findings

• A complete $\lambda$-hypersurface with polynomial area growth is isometric to $S^n(r)$, $\mathbb{R}^n$, $S^1(r)\times\mathbb{R}^{n-1}$, $S^{n-1}(r)\times\mathbb{R}$, or $S^k(\sqrt{k})\times\mathbb{R}^{n-k}$ for $2\leq k\leq n-2$ if the inequality $\left(\sqrt{S - \frac{H^2}{n}} + |\lambda|\frac{n-2}{2\sqrt{n(n-1)}}\right)^2 + \frac{1}{n}(H - \lambda)^2 \leq 1 + \frac{n\lambda^2}{4(n-1)}$ holds.
• If the inequality is violated, then the hypersurface must be isometric to one of the standard models listed above.
• When $\sup\left(\left(\sqrt{S - \frac{H^2}{n}} + |\lambda|\frac{n-2}{2\sqrt{n(n-1)}}\right)^2 + \frac{1}{n}(H - \lambda)^2\right) < 1 + \frac{n\lambda^2}{4(n-1)}$, then $S \equiv \frac{H^2}{n}$, implying the hypersurface is isometric to $S^n(r)$ with $0 < r < \sqrt{n}$ or $\mathbb{R}^n$.
• For $\lambda = 0$, the only possible isometry types are $S^n(\sqrt{n})$, $\mathbb{R}^n$, and $S^k(\sqrt{k})\times\mathbb{R}^{n-k}$.
• If $\lambda \neq 0$, the hypersurface has exactly two distinct principal curvatures, one of which is simple, and is isometric to $S^n(r)$, $\mathbb{R}^n$, $S^1(r)\times\mathbb{R}^{n-1}$, or $S^{n-1}(r)\times\mathbb{R}$.
• The second fundamental form is parallel, and the hypersurface is isoparametric, under the equality case of the key inequality, confirming the classification into standard models.