Minimal surfaces and fluctuations of membranes with nontrivial topology

A new geometric approach to the description of phase transitions and fluctu ations in membranes with nontrivial topology is proposed. The method is bas ed on the possibility of representing real membranes and vesicles, defined in the space R-3, as minimal surfaces embedded in S-3. A change in the genu s of the physical membrane corresponds to the formation of holes in the min imal surface. In the framework of mean field theory a model is constructed for a phase transition that can be characterized as the crystallization of holes in S-3. In real membranes this corresponds to a phase transition from a cubic phase to a sponge. (C) 2000 MAIK "Nauka/Interperiodica".