Geometry of Riemannian and Finsler Spaces

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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.