Subelliptic Li-Yau estimates on three dimensional model spaces
Subelliptic Li-Yau estimates on three dimensional model spaces
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发表时间：
2008-06
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通讯作者：
D. Bakry;Fabrice Baudoin;M. Bonnefont;B. Qian
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作者：
D. Bakry;Fabrice Baudoin;M. Bonnefont;B. Qian
本文描述了三维亚椭圆几何中的三种基本模型，它们分别对应于黎曼几何中的三种模型（球面、欧氏空间和双曲空间），它们分别是SU（2）、Heisenberg和SL（2）群。在这些模型上，我们证明了关于热传导方程正解的抛物型Li-Yau不等式，为此我们使用了适用于这些初等模型空间的Γ 2技巧.这里的重要特征是，尽管通常的Ricci曲率的概念是没有意义的（或者更准确地说，导致形式为-∞的Ricci曲率的界限），我们描述了一个参数ρ，它起着与Ricci曲率的下界相同的作用，并且从它推导出与黎曼几何中相同的结果，如热核上界，Sobolev不等式和直径估计。1框架和引言热核测度的估计是一个至少在过去30年里一直在深入研究的话题，见[12，8]。在为此开发的许多技术中，著名的Li-Yau抛物不等式[12]是一个非常强大的工具，它依赖于热核上梯度的黎曼几何界到Ricci曲率的下界。更确切地说，以最简单的形式，它断言，如果E是一个具有非负Ricci曲率的n维光滑黎曼流形，则如果f是热方程的任何正解，
AbstractWe describe three elementary models in three dimensional subelliptic geometry whichcorrespond to the three models of the Riemannian geometry (spheres, Euclidean spaces andHyperbolic spaces)which arerespectivelythe SU(2), Heisenbergand SL(2)groups. On thosemodels, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation.We use for that the Γ 2 techniques that we adapt to those elementary model spaces. Theimportant feature developed here is that although the usual notion of Ricci curvature ismeaningless (or more precisely leads to bounds of the form −∞ for the Ricci curvature), wedescribe a parameter ρ which plays the same rˆole as the lower bound on the Ricci curvature,and from which one deduces the same kind of results as one does in Riemannian geometry,like heat kernel upper bounds, Sobolev inequalities and diameter estimates. 1 Framework and Introduction The estimation of heat kernel measures is a topic which had been under thorough investigationfor the last thirty years at least, see [12, 8]. Among the many techniques developed for that,the famous Li-Yau parabolic inequality [12] is a very powerful tool, which relies in Riemanniangeometry bounds on the gradient on heat kernels to lower bounds on the Ricci curvature. Moreprecisely, in the simplest form, it asserts that, if E is a smooth Riemannian manifold withdimension n and non negative Ricci curvature, then if f is any positive solution of the heatequation∂