芬斯勒流形上与梯度向量场和Laplacian有关的若干重要不等式

Some Important Inequalities Related to Gradient Vector Fields and Laplacian on Finsler Manifolds

本文首先刻画了Randers流形上任一光滑函数的梯度向量场并得到了一个梯度估计.其次,本文在Ric_(N)≥K>0的条件下获得了芬斯勒Laplacian的非零特征值的一个下界估计.最后,本文在Ric_(∞)≥K>0的条件下,给出了紧致芬斯勒流形上的对数Sobolev不等式的一个全新且简单的证明.

First,we characterize the gradient vector field and obtain a gradient estimate for any smooth function on a Randers manifold.Further,we obtain an estimate of lower bound for the non-zero eigenvalues of the Finsler Laplacian under the condition that Ric_(N)≥K>0.Finally,we give a new and simple proof of the logarithmic Sobolev inequality on a compact Finsler manifold with Ric_(∞)≥K>0.