Curvature Measures in Convex and Integral Geometry

这个项目建立在S. Alesker关于赋值（即凸体空间上的有限加性函数）的工作所产生的凸几何和积分几何的决定性进展的基础上。连续和平移不变的估值空间允许自然的有限分级。根据Alesker的不可约定理，在一般线性群的作用下，每一个梯度分量都是不可约的。这对赋值空间的结构具有重要意义，并使S. Alesker发现了赋值的一系列自然代数运算，特别是交换积。此外，对凸集的限制被证明是不自然的，S. Alesker在一般光滑流形上引入了光滑赋值理论。平滑估值可能局限于平滑曲率测量，尽管不是唯一的，这是由a . Bernig和J.H.G. Fu引入的一个重要的新概念。在更多的几何术语中，光滑曲率测度是在某些奇异退化下持续存在的第二种基本形式的不变量的积分。主要的经典例子是费德勒的曲率测量，它使内在体积局部化。根据Alesker不可约定理，紧集合上一致收敛拓扑上光滑赋值的闭包正是连续赋值空间。这个项目的一个基本问题和主要目标是类似地完成光滑曲率测度的空间，并通过一系列不可避免的属性来表征完成的要素。该项目将追求这样一个想法，即许多关于平移不变估值的定理应该在一些不同的背景下有一个对应的曲率测量。特别地，我们将研究一般线性群的作用。平滑的曲率测量也为内在体积开辟了新的视角。Alesker已经证明，凸几何中的内禀体积概念与每个黎曼流形联系在一起的是一个有限维的赋值代数，即Lipschitz-Killing代数。光滑曲率测度的空间自然是光滑赋值上的一个模，这一事实开辟了一种新的显然非常自然的方法：通过a. Bernig， J.H.G. Fu和S. Solanes的一个猜想，光滑赋值使子空间角曲率测度保持不变（相对于Alesker积），它由一个简单的几何性质来区分，当且仅当它是Lipschitz-Killing代数的一个元素。最近我们建立了这个猜想的“如果”部分。在这个项目中，我们的主要目标之一是探索这个猜想的“only if部分”。

This project builds on decisive progress in Convex and Integral Geometry arising from the work of S. Alesker on valuations, i.e., finitely additive functions on the space of convex bodies. The space of continuous and translation-invariant valuations admits a natural finite grading. By Alesker's Irreducibility Theorem, each graded component is irreducible under the action of the general linear group. This has strong implications for the structure of the space of valuations and led S. Alesker to discover a range of natural algebraic operations on valuations, in particular a commutative product. Furthermore the restriction to the convex setting turned out to be unnatural, and S. Alesker has introduced a theory of smooth valuations on general smooth manifolds.Smooth valuations may be localized, albeit not uniquely, to smooth curvature measures, a crucial new concept introduced by A. Bernig and J.H.G. Fu. In more geometric terms, a smooth curvature measure is an integral of invariants of the second fundamental form that persists under certain singular degenerations. The prime classical examples are Federer's curvature measures, which localize the intrinsic volumes. By Alesker's Irreducibility Theorem, the closure of smooth valuations in the topology of uniform convergence on compact sets is precisely the space of continuous valuations. A fundamental issue and main objective in this project is to similarly complete the space of smooth curvature measures and to characterize the elements of the completion by a short list of inevitable properties. The project will persue the idea that many theorems on translation-invariant valuations should have a counterpart for curvature measures in a few different contexts. In particular, the action of the general linear group will be investigated. Smooth curvature measures also open up a new perspective on the intrinsic volumes. Alesker has shown that the notion of intrinsic volumes from Convex Geometry associates to each Riemannian manifold a finite-dimensional algebra of valuations, the Lipschitz-Killing algebra. The fact that the space of smooth curvature measures is naturally a module over smooth valuations opens up a new and evidently very natural approach: By a conjecture of A. Bernig, J.H.G. Fu, and S. Solanes, a smooth valuation leaves invariant (with respect to the Alesker product) the subspace angular curvature measures, which is distinguished by a simple geometric property, if and only if it is an element of the Lipschitz-Killing algebra. Recently we have established the ``if part'' of this conjecture. One of our primary goals in this project is to explore the ``only if part'' of this conjecture.