Superfícies de Weingarten lineares em R3

Abstract

In this dissertation, we will study some properties of linear Weingarten surfaces in R3. These are immersions of an abstract surface S in R3, for which there are three real numbers a, b and c, not all null, satisfying 2aH(P) + bK(P) = c for all P 2 S, with H being the mean curvature and K the Gaussian curvature of S, respectively. We will give an estimate for the height of an elliptical linear Weingaten surface (a2 + bc > 0), compact, with respect to a plane. We will also give an estimate for 2aH + bK on a compact linear Weingarten surface and a compact plot with a convex planar edge. Also, let's prove the following result: Let S be a closed topological disk and : S −! R3 a linear Weingarten embedding satisfying a2+bc >0. If the edge image of S, (@S), is a line of curvature then (S) is contained in a plane or a sphere. To prove this result, we will need the calculation of the Laplacians of two functions, in relation to a special Riemannian metric (Proposition 2.2) .