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Search Results: 1 - 10 of 46976 matches for " Liang Cheng "
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On the extension to mean curvature flow in lower dimension
Cheng Liang
Mathematics , 2013,
Abstract: In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for $T<\infty$, then the flow can be smoothly extended over time $T$.
An asymptotic property of Huisken's functional on minimal submanifolds of Euclidean space
Liang Cheng
Mathematics , 2012,
Abstract: In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.
A mean value inequality for the generalized self-expander type submanifolds and its application
Liang Cheng
Mathematics , 2015,
Abstract: In this paper we get a version of mean value inequality for generalized self-expander type submanifolds in Euclidean space. As the application, we prove that if mean curvature flow $M(t)$ on the self-expander in Euclidean space subconverges to an $n$-rectifiable varifold $T$ in weak sense for $t$ goes to the singular time, then $T$ must be the cone.
Relationship between microstate number and available entransy
XueTao Cheng,XinGang Liang
Chinese Science Bulletin , 2012, DOI: 10.1007/s11434-012-5152-9
Abstract: Based on the relationship between entransy and microstate number, we discuss the variations of the available transport entransy, the unavailable transport entransy, the available conversion entransy and the unavailable conversion entransy with the microstate number. We focus on physical processes in which heat is used for heating/cooling or doing work. When heat is transported for heating or cooling, the available transport entransy increases if the increase in microstate number is due to the increase in internal energy of the system, and decreases if the increase in microstate number is due to spontaneous heat transfer. When heat is used to do work, both the available conversion entransy and the unavailable conversion entransy increase if the increase in microstate number relates to the growth in internal energy of the system. The available conversion entransy decreases and the unavailable conversion entransy increases if the increase in microstate number results from spontaneous heat transfer.
Curvature tensor under the complete non-compact Ricci Flow
Li Ma,Liang Cheng
Mathematics , 2008,
Abstract: We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\times [0,T)$. Some other results are also presented.
Gradient estimate for the Poisson equation and the non-homogeneous heat equation on compact Riemannian manifolds
Li Ma,Liang Cheng
Mathematics , 2009,
Abstract: In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
Non-local heat flows and gradient estimates on closed manifolds
Li Ma,Liang Cheng
Mathematics , 2009,
Abstract: In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.
A non-local area preserving curve flow
Li Ma,Liang Cheng
Mathematics , 2009,
Abstract: In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in C^{\infty} sense as time goes into infinity.
On the weighted forward reduced Entropy of Ricci flow
Liang Cheng,Anqiang Zhu
Mathematics , 2010,
Abstract: In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4\pi)^{\frac{n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.
Curvature tensor under the Ricci-Harmonic flow
Anqiang Zhu,Liang Cheng
Mathematics , 2011,
Abstract: We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.
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