Geometry of Riemannian and Finsler Spaces

摘要提案：DMS-9803129首席研究员：Dmitry Burago提案的主要主题（大尺度几何和台球系统）属于几何和动力系统。D. Burago继续他（与S. Ivanov联合）对周期度量的大尺度几何的研究，提出研究Finsler环面的渐近体积增长；一个相关的问题是分析Banach空间中仿射子空间的面积最小化性质。这有助于理解环面上拉格朗日系统共轭点的缺失是如何反映在其测地线流动动力学中的。D. Burago还建议继续对奥布里-马瑟理论的几何结论进行高维类比分析。试图进一步理解biLipschitz等价和准等距之间的关系，很自然地要考虑密度参数（由提议者与B. Kleiner共同开发）不起作用的情况；最引人注目的例子包括彭罗斯平铺和均匀格。继续他（与S. Ferleger， A. Kononenko联合）对半分散台球系统的研究，D. Burago计划调查它们的拓扑熵是否可以是无限的。将奇异几何应用于台球理论的另一个问题是构造CAT(0)发展空间，其测地线表示所有的台球轨迹。作者的导师E. Johnson运用作者的方法证明了曲率有限变化的完全曲面的无界性，以证明嵌入平面类的稳定性。该提案的主要主题有非常明确的物理类比。周期度量的大尺度几何特性可以解释为周期介质的全局特性，周期介质由相同微观模式的副本以规则的方式重复组成，如在晶体中。特别是，周期尺度的地理设计流的动力学性质反映了光或辐射如何在这种介质中传播。由提议者和他的合作者开发的技术允许我们解决这个领域的许多开放问题，对于最重要的情况（二次拉格朗日量）；总的来说，这一系列的问题仍然没有解决。提议者对台球系统理论的研究始于一个可以追溯到玻尔兹曼的问题：在给定的时间间隔内，在一个由几个球弹性碰撞的系统（气体模型）中，能否给出最大碰撞次数的上界？单位时间内的碰撞次数在热力学中起着重要的作用。令人惊讶的是，这个问题已经通过建立与奇异几何的联系解决了；反过来，这种联系又引出了几何和动力学两方面的新问题。

Abstract Proposal: DMS-9803129 Principal Investigator: Dmitry Burago The main topics of the proposal (large-scale geometry and billiards systems) belong to both geometry and dynamical systems. Continuing his (joint with S. Ivanov) research in the large-scale geometry of periodic metrics, D. Burago proposes to study the asymptotic volume growth for Finsler tori; a related problem is to analyze area-minimizing properties of affine subspaces in Banach spaces. This may help to understand how the absence of conjugate points for a Lagrangian system on a torus is reflected in the dynamics of its geodesic flow. D. Burago also proposes to continue his analysis of higher dimensional analogs for geometric conclusions of Aubry-Mather theory. Trying to further understand the relationship between biLipschitz equivalence and quasi-isometries, it is natural to consider such cases where density arguments (developed by the proposer jointly with B. Kleiner) do not work; the most striking of such cases include Penrose tilings and uniform lattices. Continuing his (joint with S. Ferleger, A. Kononenko) study of semi-dispersing billiard systems, D. Burago plans to investigate if their topological entropy can be infinite. Another problem which arose from the proposer's method of applying singular geometry to billiard theory is constructing CAT(0) development spaces whose geodesics represent all billiard trajectories. E. Johnson, the proposer's advisee, works on applying the proposer's method to prove the unboundness for complete surfaces of finite variation of curvature to show stability of the class of embedded flat surfaces. The main subjects of the proposal have very clear physical analogs. Large-scale geometric properties of periodic metrics can be interpreted as global properties of a periodic medium consisting of copies of the same microscopic pattern repeated in a regular fashion, as in crystals. In particular, dynamical properties of geo desic flows for periodic metrics reflect how the light or radiation spreads in such media. The technique developed by the proposer and his collaborators allows us to solve many of the open problems in this area for the most important case (quadratic Lagrangians); in general, this circle of problems remains wide open. The proposer's research in the theory of billiard systems started from a problem that goes back to Boltzmann: can one give an upper bound on the maximum number of collisions in a given time interval in a system of several balls colliding elastically (gas model)? The number of collisions per time unit plays important role in thermodynamics. Surprisingly, this problem has been solved by establishing a connection with singular geometry; in its turn, this connection led to new intriguing problems of both geometric and dynamical origin.