Analysis on Riemann surfaces and manifolds to gether with contributions to physics and engineeing

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We used the function-theoretic methods to obtain a new insight into the various fluid-dynamical phenomena on surfaces and manifolds. Specifically, we have been interested in Rankine ovoids on plane and Riemann surfaces. We use the function-theoretic technique (initiated by the head investigator) to study the phenomenon physically as well as mathematically. The physical investigation is based on the notions of sink and source, vortex, energy, moment and so on, while the mathematical investigation is based on our own results which have been so far obtained. Of the most importance among them is the study of compact Riemann surfaces into which a prescribed noncompact Riemann surface can be conformally embedded. We have proved that any Rankine ovoid can be realized by a univalent meromorphic function on a simply connected plane domain and that the area of the Rankine ovoid is surprisingly close to the possible maximum area which will be attained in a certain family of competing functions. The result was announced at the International Conference "Computational Methods and Function Theory (CMFT'97)" held in Nicosia, Cyprus, and will be included in the Proceedings.