[Paper Review] Rigidity theorems of the space-like $λ$-hypersurfaces in the Lorentzian space $\mathbb R^{n+1}_1$

Xingxiao Li, Xiufen Chang|arXiv (Cornell University)|Nov 10, 2015

Geometric Analysis and Curvature Flows5 references3 citations

TL;DR

This paper establishes rigidity theorems for complete space-like $\lambda$-hypersurfaces in the Lorentzian space $\mathbb{R}^{n+1}_1$, extending results from Euclidean space by introducing a generalized $\mathcal{L}$-operator and analyzing weighted $L^2$ integrals of curvature quantities. The key result shows that such hypersurfaces are either totally umbilical (isometric to $\mathbb{H}^n(c)$ or $\mathbb{R}^n$) or satisfy a strict inequality involving the squared norm of the second fundamental form and mean curvature, implying non-trivial geometric constraints.

ABSTRACT

In this paper, we study complete space-like $λ$-hypersurfaces in the Lorentzian space $\mathbb R^{n+1}_1$. As the result, we prove some rigidity theorems for these hypersurfaces including the complete space-like self-shrinkers in $\bbr^{n+1}_1$.

Motivation & Objective

• To extend rigidity theorems for $\lambda$-hypersurfaces from Euclidean space to the Lorentzian setting $\mathbb{R}^{n+1}_1$.
• To define and analyze $\lambda$-hypersurfaces and self-shrinkers in the Lorentzian space using a generalized $\mathcal{L}$-operator.
• To establish conditions under which complete space-like $\lambda$-hypersurfaces must be totally umbilical or satisfy a geometric inequality.
• To generalize prior Bernstein-type and rigidity results for constant mean curvature and self-shrinker hypersurfaces to the Lorentzian context.

Proposed method

• Introduce a generalized $\mathcal{L}$-operator adapted to the Lorentzian metric, defined using the Hessian and gradient of curvature quantities.
• Use weighted $L^2$ integrals with exponential weight $e^{-\frac{\epsilon a\langle x,x\rangle}{2}}$ to control curvature decay and apply integration by parts.
• Apply a generalized Bochner-type formula to derive an expression for $\frac{1}{2}\mathcal{\tilde{L}}B$, where $B = S - \frac{H^2}{n}$, involving curvature terms and the $\lambda$-condition.
• Use the inequality $|f_3| \leq \frac{n-2}{\sqrt{n(n-1)}} B^{3/2}$ to bound the cubic term in the curvature expression.
• Apply a weighted maximum principle via Corollary 2.4 to deduce that if the weighted integral of the curvature expression is non-positive, then curvature terms must vanish or satisfy a strict inequality.
• Use the isoparametric property derived from parallel second fundamental form to classify the hypersurface as a product of hyperbolic and Euclidean spaces, leading to a contradiction unless $B \equiv 0$ or the inequality in (1.6) holds.

Experimental results

Research questions

• RQ1 Under what conditions are complete space-like $\lambda$-hypersurfaces in $\mathbb{R}^{n+1}_1$ rigid, i.e., must they be totally umbilical or satisfy a geometric inequality?
• RQ2 How does the $\mathcal{L}$-operator in the Lorentzian setting behave under weighted $L^2$ integration, and what does it reveal about curvature decay?
• RQ3 Can the rigidity results for $\lambda$-hypersurfaces in Euclidean space be extended to the Lorentzian case with non-constant weight functions such as $s = \langle x,x\rangle$?
• RQ4 What role does the sign of $\langle x,x\rangle$ play in the classification of $\lambda$-hypersurfaces in $\mathbb{R}^{n+1}_1$?
• RQ5 Is it possible for a complete space-like $\lambda$-hypersurface in $\mathbb{R}^{n+1}_1$ to be isoparametric with two distinct principal curvatures and non-constant $\langle x,x\rangle$, while satisfying the $\lambda$-hypersurface equation?

Key findings

• Complete space-like $\lambda$-hypersurfaces in $\mathbb{R}^{n+1}_1$ with weight $s = \epsilon a$ and polynomial area growth are either totally umbilical (isometric to $\mathbb{H}^n(c)$ or $\mathbb{R}^n$) or satisfy the inequality in (1.6) at some point.
• The squared norm $B = S - \frac{H^2}{n}$ of the traceless second fundamental form must vanish identically or violate the inequality $\left(\sqrt{B} - |\lambda|\frac{n-2}{2\sqrt{n(n-1)}}\right)^2 + \frac{1}{n}(H - \lambda)^2 - \frac{n\lambda^2}{4(n-1)} \geq 0$ at some point.
• If $B \not\equiv 0$, then the second fundamental form is parallel, implying the hypersurface is isoparametric with exactly two distinct principal curvatures, one of which is simple.
• Such isoparametric hypersurfaces in $\mathbb{R}^{n+1}_1$ are classified as products $\mathbb{H}^{n-1}(c) \times \mathbb{R}^1$ or $\mathbb{H}^1(c) \times \mathbb{R}^{n-1}$, but these cannot satisfy the $\lambda$-hypersurface condition with $s = \langle x,x\rangle$ due to non-constant $\langle x,x\rangle$, leading to a contradiction.
• The analysis shows that the only possible complete $\lambda$-hypersurfaces with $s = \langle x,x\rangle$ are those for which $B \equiv 0$ or the inequality (1.6) holds at some point, proving rigidity under the given integral condition.
• The weighted $L^2$ integrability condition (1.2) ensures that the curvature terms vanish or the geometric inequality is satisfied, leading to classification via the $\mathcal{L}$-operator and integration techniques.