For an arbitrary Riemannian 3-manifold \(N^3\), a \(C^{\infty}\)-map \(f:M^2\to N^3\) from a 2-manifold \(M^2\) is called a (wave) front if \(f\) lifts to a smooth immersed section \(L_f:M^2\to T_1N^3\) of the unit tangent vector bundle \(T_1N^3\) such that \(df(X)\) is perpendicular to \(L_f(p)\) for all \(X\in T_pM^2\) and \(p\in M^2\). The map \(f\) is called a \(p\)-front if for each \(p\in M^2\), there is a neighborhood \(U\) of \(p\) such that the restriction \(f|_U\) is a front. Following the discovery of [J. A. Gálvez, A. Martínez and F. Milán, Math. Ann. 316, No. 3, 419–435 (2000; Zbl 1003.53047)], the first, third and fourth authors gave a framework for complete flat fronts with singularities in the hyperbolic space \(H^3\).
In this paper, the authors broaden the notion of completeness to weak completeness, and of front to \(p\)-front, so that the new framework and results can apply to a more general class of flat surfaces. This more general class contains the caustics of flat fronts – shown also to be flat by Roitman (who gave a holomorphic representation formula for them) – which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only \(p\)-fronts. By using the new framework, the authors obtain characterizations for caustics.