Sur les scindements de Heegaard du tore $T\sp 3$

Abstract

We prove that there is, up to isotopy, a unique Heegaard splitting of a given genus g > 3 in the 3-torus Γ 3 . Using Meek's results, this classification theorem gives that two minimal surfaces of a given genus g > 3 in a flat torus are isotopic. This implies, in particular, the topological uniqueness of triply periodic minimal surfaces in R .

Heegaard Genus of the Connected Sum of M-small Knots

“…Thus, we may assume that g(X) < g(X I ) + g(X J ). In that case, Corollary 5.4 implies that g(X (1) I ) = g(X I ). Our second tool, the Hopf-Haken Annulus Theorem (Theorem 6.3), studies minimal genus Heegaard splitting of X (1) I .…”

“…In that case, Corollary 5.4 implies that g(X (1) I ) = g(X I ). Our second tool, the Hopf-Haken Annulus Theorem (Theorem 6.3), studies minimal genus Heegaard splitting of X (1) I . Denote the boundary of X admits an essential annulus (say A) connecting ∂X I to T , so that A ∩ ∂X I is a meridian of X I and A ∩ T is a longitude of T .…”

“…2 , where F (1) and F (2) are non-empty and mutually homeomorphic. Let M be a manifold obtained from M (1) and M (2) by identifying F (1) and F (2) .…”

“…Let M be a manifold obtained from M (1) and M (2) by identifying F (1) and F (2) . Then F denotes the image of F (1) (= the image of F (2) ) in M. Suppose that we are given Heegaard splittings…”

“…Let M (1) and M (2) be compact, orientable 3-manifolds with boundary, together with partitions of the boundary components:…”