tailieunhanh - Đề tài " Growth of the number of simple closed geodesics on hyperbolic surfaces "

In this paper, we study the growth of sX(L), the number of simple closed geodesics of length ≤ L on a complete hyperbolic surface X of finite area. We also study the frequencies of different types of simple closed geodesics on X and their relationship with the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. | Annals of Mathematics Growth of the number of simple closed geodesics on hyperbolic surfaces By Maryam Mirzakhani Annals of Mathematics 168 2008 97 125 Growth of the number of simple closed geodesics on hyperbolic surfaces By Maryam Mirzakhani Contents 1. Introduction 2. Background material 3. Counting integral multi-curves 4. Integration over the moduli space of hyperbolic surfaces 5. Counting curves and Weil-Petersson volumes 6. Counting different types of simple closed curves 1. Introduction In this paper we study the growth of sx L the number of simple closed geodesics of length L on a complete hyperbolic surface X of finite area. We also study the frequencies of different types of simple closed geodesics on X and their relationship with the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Simple closed geodesics. Let cX L be the number of primitive closed geodesics of length L on X. The problem of understanding the asymptotics of cX L has been investigated intensively. Due to work of Delsarte Huber and Selberg it is known that cx L eL L as L ữQ. By this result the asymptotic growth of cx L is independent of the genus of X. See Bus and the references within for more details and related results. Similar statements hold for the growth of the number of closed geodesics on negatively curved compact manifolds Ma . However very few closed geodesics are simple BS2 and it is hard to discern them in n1 X BS1 . Counting problems. Let Mg n be the moduli space of complete hyperbolic Riemann surfaces of genus g with n cusps. Fix X G Mgn. To understand the 98 MARYAM MIRZAKHANI growth of sx L it proves fruitful to study different types of simple closed geodesics on X separately. Let Sgn be a closed surface of genus g with n boundary components. The mapping class group Modg n acts naturally on the set of isotopy classes of simple closed curves on Sg n. Every isotopy class of a simple closed curve contains a unique simple closed geodesic on X. Two simple .