Entropy and topology of two-dimensional extreme black holes

Through direct thermodynamic calculations we have shown that different classical entropies of two-dimensional extreme black holes appear due to two different treatments, namely Hawking's treatment and Zaslavskii's treatment. Geometrical and topological properties corresponding to these different treatments are investigated. Quantum entropies of the scalar fields on the backgrounds of these black holes concerning different treatments are also exhibited. Different results of entropy and geometry lead us to argue that there are two kinds of extreme black holes in nature. An explanation of black hole phase transition has also been given from the quantum point of view. [S0556-2821(99)03908-9].