On Reeb graphs induced from smooth functions on closed or open surfaces

Naoki Kitazawa

Published 2019-08-12Version 1

For a smooth function on a smooth manifold of a suitable class, the space of all the connected components of inverse images is the graph and called the Reeb graph. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions not so hard to handle: the global singularity theory. In this paper, we attack the following natural problem: can we construct a smooth function with good geometric properties inducing a given graph as the Reeb graph. This problem has been essentially launched by Sharko in 2000s and various answers have been given by Masumoto, Michalak, Saeki etc.. Recently the author has set a new explicit problem and given an answer. In the studies before the result of the author, considered functions are smooth functions on closed surfaces or Morse functions such that inverse images of regular values are disjoint unions of standard spheres: well-known most fundamental Morse functions with just $2$ singular points, characterizing sphere topologically, are examples of such functions. On the other hand, the author has succeed in construction of a smooth function on a $3$-dimensional closed, connected and orientable manifold inducing a given graph as the Reeb graph such that inverse images of regular values are desired. Based on the studies, especially on the result and method of the author, with several new ideas, we will consider smooth functions on surfaces which may be non-closed and give an answer.