ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS
ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS
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DOI：
10.2140/pjm.2013.264.61
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发表时间：
2013-05
影响因子：
0.6
通讯作者：
Hongxin Guo;Robert Philipowski;Anton Thalmaier
中科院分区：
文献类型：
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作者：
Hongxin Guo;Robert Philipowski;Anton Thalmaier
郭洪欣，罗伯特·菲利普斯基，安东·塔尔迈尔摘要。本文确定了一般发展流形上共轭热方程的经典Boltzmann-Shannon熵的前两阶导数。基于Boltzmann-Shannon熵的二阶导数，我们构造了抽象几何流中的Perelman F熵和W熵.当萨蒂斯一个技术条件时，熵的单调性成立.这个条件在具有非负Ricci曲率的静态黎曼流形上萨蒂斯，对于汉密尔顿的Ricci流形，List的推广Ricci流形，当周围空间具有非负截面曲率时，Müller的Ricci算子与调和映射算子和Lorentz平均曲率算子耦合，在最低特征值随时间变化的额外假设下，在抽象背景下，我们导出了具有位势的Laplace-Beltrami算子最低本征值的演化的显式公式。
HONGXIN GUO, ROBERT PHILIPOWSKI, AND ANTON THALMAIERAbstract. In this note we determine the first two derivatives of the clas-sical Boltzmann-Shannon entropy of the conjugate heat equation on generalevolving manifolds. Based on the second derivative of the Boltzmann-Shannonentropy, we construct Perelman’s F and W entropy in abstract geometric flows.Monotonicity of the entropies holds when a technical condition is satisfied.This condition is satisfied on static Riemannian manifolds with nonnegativeRicci curvature, for Hamilton’s Ricci flow, List’s extended Ricci flow, Mu¨ller’sRicci flow coupled with harmonic map flow and Lorentzian mean curvatureflow when the ambient space has nonnegative sectional curvature.Under the extra assumption that the lowest eigenvalue is differentiablealong time, we derive an explicit formula for the evolution of the lowest eigen-value of the Laplace-Beltrami operator with potential in the abstract setting.