Complete space-like self-expanders in the Minkovski space
Zhi Li, Guoxin Wei
TL;DR
We address the classification of complete space-like self-expanders in the Minkowski space $\\mathbb{R}^{n+1}_{1}$ under geometric constraints. The authors develop a framework based on the Bakry-Emery Laplacian $\\mathcal{L}=\\Delta+\\langle x^{\\top},\\nabla\\rangle$ and apply an $\\mathcal{L}$-version of the Omori-Yau maximum principle to derive drift-diffusion identities for $H$ and $S$ and to analyze the constancy of $S$. The main results show that, when $S$ is constant and $\\inf H^{2}>0$, one must have $S=1$ and $x(M^{n})$ is either the hyperbolic space $\\mathbb{H}^{n}(\\sqrt{n})$ or a hyperbolic cylinder $\\mathbb{H}^{k}(\\sqrt{k})\\times\\mathbb{R}^{n-k}$; in dimension two, complete space-like self-expanders with constant $S$ are restricted to a space-like affine plane through the origin, $\\mathbb{H}^{1}(1)\\times\\mathbb{R}$, or $\\mathbb{H}^{2}(\\sqrt{2})$. The Euclidean analogue in $\\mathbb{R}^{3}$ yields only the plane, illustrating robustness of the rigidity phenomenon. These results extend rigidity phenomena for self-expanders to Lorentzian ambient spaces and illuminate singularity models for mean curvature flow in Minkowski space.
Abstract
It is our purpose to study complete space-like self-expanders in the Minkovski space. By use of maximum principle of Omori-Yau type, we can obtain the rigidity theorems on $n$-dimensional complete space-like self-expanders in the Minkovski space $\mathbb R^{n+1}_{1}$. For complete space-like self-expanders of dimension $2$, we give a classification of them under assumption of constant squared norm of the second fundamental form.