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Let 5 be a two dimensional minimal surface in R, and let P c R be a thin curved rectangle whose two short sides lie along dS and that is otherwise disjoint from S. More generally S can be an m-dimensional minimal surface and P can be a set homeomorphic to [0,1] x i?TM. Typically S will have two connected components, and P will join one to the other. The bridge principle for minimal surfaces is the principle that it should usually be possible to deform S U P slightly to make a minimal surface with boundary d(S U P). In a previous paper, we showed that it is possible provided that S is smooth and strictly stable, that P is sufficienty thin, and that, at each end of P, the angle between P and S is strictly between 0 and 27r. ("Strictly stable" means "stable and having no nonzero jacobi fields that vanish on the boundary" or, equivantly, "having index 0 and nullity 0 as a critical point for the area functional".) In §1 of this paper we extend that result to all unstable smooth surfaces S that have nullity 0. As a corollary we prove that a certain simple closed curve in dB that is smooth except at one point has the following property. For every genus g < oo, every area a in some interval [L, +oo], and every index i < oo, there exist uncountably many (namely 2^°) embedded minimal surfaces with genus g, area a, and index L. In §2 we give two examples to show that the nullity 0 assumption is necessary. First, we show that if M is a catenoid of nullity 1 with boundary in dB, if iV is a minimal surface with boundary in one of the simply connected

The bridge principle for unstable and for singular minimal surfaces