arXiv subjects
Publications and source records attributed to Hengyu Zhou.
We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial curve is a graph over $S^1$, then its curve shortening flow exists for all times and finally converges to a geodesic closed curve.
We present the Levy's distributional property for symmetric Levy processes with generating triplet $(0, 0,ν)$ or $(σ>0, γ, ν)$ where $ν$ is a symmetric measure on $R\backslash\{0\}$. This generalizes the classical Levy's theorem about Brownian motions with drift.
Let $M$ be a closed Riemannian manifold with a parallel 1-form $Ω$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $Ω(T)\geq 0$ on the initial curve $\co$. Here $T$ is the unit tangent vector on $\co$. The other one is about the convergence. It says that in a closed {\Rm} $\tilde{M}$, assume the curve shortening flow $\ct$ exists for all $t\in[0,\infty)$ and its length converges to a positive limit, then $ \lim\limits_{t\rightarrow\infty}max_{\ct}|\nabla^{m}A|^{2}=0$ for all $m=0,1,...$. Here $A$ denotes the second fundamental form of $\ct$ in $\tilde{M}$.