Survey of apparent contours of stable maps between surfaces

This is a survey paper about studies of the simplest shape of the apparent contour for stable maps between surfaces. Such studies first appeared in [10] then in [1], [3], [6], [20], [22]. Let M be a connected and closed surface, N a connected surface. For a stable map phi: M -> N, denote by c(phi), n(phi) and i(phi) the numbers of cusps, nodes and singular set components of phi, respectively. For a C-infinity map phi(0) : M -> S-2 into the sphere, we study the minimal pair (i, c + n) and triples (i, c, n), (c, i, n), (n, c, i) and (i, n, c) among stable maps M -> S-2 homotopic to phi(0) with respect to the lexicographic order.