In this work, we discuss several methods that can solve topological changes during the Lagrangian surface evolution. The Lagrangian approach, contrarily to level-set methods, solves the surface evolution problem in a two-dimensional setting, but the processes of cutting and gluing the evolving surfaces are non-trivial. We present examples of mean curvature flow and an evolution by a constant normal speed. We made approximation of the mean curvature flow by a finite volume method -- the so called cotangent scheme. Then we suggest two methods that solve the most common topological changes."