Discrete Koenigs nets and finite Laplace sequences

Abstract

Q-nets are maps from the square grid to projective space that have planar faces. We consider the Laplace sequences of Q-nets, which are determined by iterating a discrete time dynamics called Laplace transformations. In general, the Laplace sequences are bi-infinite. However, there are special cases in which a Laplace transform degenerates to a curve. In these cases we say that the sequences terminates. In this paper, we consider two special cases of Q-nets which are both called (discrete) Koenigs nets. For these Koenigs nets we show that if the sequence terminates, then the sequence is finite. More specifically, we show that if the Laplace transform is Laplace degenerate (or Goursat degenerate) after m steps in one direction, then it is Laplace degenerate after m + 1 (or m + 2) steps in the other direction.