| The Heisenberg group H1 together with its standard sub-Riemannian metric g0 is considered as the boundary of a Riemannian manifold, C2 +, which is isomorphic to the direct product H 1 x R+. The restriction, gu, of the Riemannian metric of C 2+ to each hypersurface H 1 x {lcub}u{rcub} degenrates to g 0 as u → 0. This theses has three parts. In the first part we study the properties of geodesics in C 2+ and on the hypersufaces H 1 x {lcub}u{rcub} in detail. These properties are strongly related to those of the Heisenberg group. In the second part, we study the Riemannian Laplace-Beltrami operator L on C 2+, calculate the heat kernel and Green's function for L, and give global and small time estimates of the heat kernel. We also calculate the heat kernel and the Green's function for the restriction of L to the hypersurface H 1 x {lcub}u{rcub}. We see that they converge to the heat kernel and the Green's function on the boundary, respectively. In the last part of this thesis, we consider some invariant surfaces with constant mean curvature in (H1, gu). These surfaces of (H1, gu) have very nice limits as u → 0. We then define the mean curvature of a hypersurface in (H1, g 0) to be the limit of its mean curvature in ( H1, gu). We show that in a more general case, this definition is appropriate. |
