Funciones de Morse minimales en el espacio dodecaédrico de Poincaré, vía la Ecuación del Calor

Abstract

Let (M,g) be a compact, connected, without boundary riemannian manifold that is homogeneous, i.e. each pair of points p, q 2M have isometric neighborhoods -- This thesis is a another step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to @f/@t = gf, f (·, 0) = f0 is such that for sufficiently large t, f (·, 0) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M -- In this thesis we show that for the Poincaré dodecahedral space this seems to hold if one allows a generic small perturbation of the metric -- Concretely, we consider an approximation of the spherical Poincaré dodecahedral space by a suitably weighted graph, calculate the eigenvalues and eigenvectors of its laplacian oparator, and study the critical point structure of eigenvectors of some of the first nonzero eigenvalues, and observe that they have the least possible number of critical points