几何流和几何不等式是几何分析中的两个核心问题。本项目将对各向异性曲率流和Alexandrov-Fenchel不等式展开研究。各向异性曲率流可以看成是有限维赋范空间中超曲面的曲率流，它来源于带有不同界面结构的多相热力学中界面的演化的物理背景，也与凸体理论中的混合体积密切相关。Alexandrov-Fenchel不等式是凸体理论中的核心不等式之一。本项目拟研究如下问题：.1.完全非线性各向异性曲率流与逆各向异性曲率流的存在收敛性；.2.欧式空间星形k凸超曲面的各向异性Alexandrov-Fenchel不等式；.3.非欧空间形式中凸超曲面的Alexandrov-Fenchel不等式。.本项目所使用方法主要是完全非线性抛物型偏微分方程理论以及利用几何流的收敛性证明几何不等式的方法。各向异性将会造成技术上的困难，尤其出现在偏微分方程的先验估计。

Geometric flow and geometric inequality are two central subjects in geometric analysis. This project is concerned with anisotropic curvature flow and Alexandrov-Fenchel inequality. The anisotropic curvature flow can be viewed as curvature flow of hypersurfaces in finite dimensional normed space. It arises from the physical model of evolution of interfaces in multiphase thermomechanics with interfacial structure. Also it is related closely to the concept of mixed volume in the theory of convex body. On the other hand, Alexandrov-Fenchel inequality is one of the most important inequality in the theory of convex body. We will mainly consider the following three problems:.1. existence and convergence of fully nonlinear anisotropic curvature flow and inverse type anisotropic curvature flow;.2. anisotropic Alexandrov-Fenchel inequality for star-shaped, k-convex hypersurfaces in the Euclidean space;.3. Alexandrov-Fenchel inequality for convex hypersurfaces in the non-Euclidean space forms..The method which will be used in this project is mainly fully nonlinear parabolic PDE theory and flow approach to geometric inequality. The anisotropy will cause technical difficulties, particularly in the a priori estimates of PDEs.