Invariants and balancing conditions for singularities of solutions of geometric variational problems

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We analize the collapse of n-end catenoids of genus O, a significant class of minimal surfaces in the Euclidian 3-space, by means of the relative weights of end-pairs of the surfaces. On the other hand, we generalize the formulation of the existence condition of n-end catenoids in the case of genus O to the case of genus 1.Moreover, we also generalize the formulation to the case of maximal surfaces in the Lorentzian 3-space, and furthermore, analize the correspondence between the symmetry of maximal surfaces and the singular set.