tailieunhanh - Đề tài " Random k-surfaces "

Invariant measures for the geodesic flow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and well-studied subject. This paper continues an investigation into a 2-dimensional analog of this flow for a 3-manifold N . Namely, the article discusses 2-dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1, surfaces which are called k-surfaces. The “2-dimensional” analog of the unit tangent bundle with the geodesic flow is a “space of pointed k-surfaces”, which can be considered as the space of germs of complete k-surfaces passing. | Annals of Mathematics Random k-surfaces By Francois Labourie Annals of Mathematics 161 2005 105 140 Random -surfaces By Francois Labourie Abstract Invariant measures for the geodesic flow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and well-studied subject. This paper continues an investigation into a 2-dimensional analog of this flow for a 3-manifold N. Namely the article discusses 2-dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1 surfaces which are called k-surfaces. The 2-dimensional analog of the unit tangent bundle with the geodesic flow is a space of pointed k-surfaces which can be considered as the space of germs of complete k-surfaces passing through points of N. Analogous to the 1-dimensional lamination given by the geodesic flow this space has a 2-dimensional lamination. An earlier work 1 was concerned with some topological properties of chaotic type of this lamination while this present paper concentrates on ergodic properties of this object. The main result is the construction of infinitely many mutually singular transversal measures ergodic and of full support. The novel feature compared with the geodesic flow is that most of the leaves have exponential growth. 1. Introduction We associated in 1 a compact space laminated by 2-dimensional leaves to every compact 3-manifold N with curvature less than -1. Considered as a dynamical system its properties generalise those of the geodesic flow. In this introduction I will just sketch the construction of this space and will be more precise in Section 2. Let k E 0 1 . A k-surface is an immersed surface in N such that the product of the principal curvatures is k. If N has constant curvature K a k-surface has curvature K k. Analytically k-surfaces are described by elliptic equations. L auteur remercie 1 Institut Universitaire de France. 106 FRANCOIS LABOURIE When dealing with ordinary differential solutions one is