DIRICHLET FINITE SOLUTIONS OF Au = Pu, AND CLASSIFICATION OF RIEMANN SURFACES
نویسنده
چکیده
1. Problem and its background. Consider a C differential P(z)dxdy (z = x+iy) on an open Riemann surface R with P(z)*t0. We denote by PX(R) the set of C solutions on R of the elliptic equation Au=Pu, or more precisely, of d*du(z)—u(z)P(z)dxdy, with a certain property X. For P = 0 we use the traditional notation HX instead of OX. Let Opx be the set of pairs (R} P) such that PX(R) reduces to constants. Instead of (i£, P)(E.Opx we simply write R&OPX if P is well understood. As for X we let B stand for boundedness, D for the finiteness of the Dirichlet integral DR(U) ~JR du/\*du, and E for the finiteness of the energy integral ER(U) =DR(U) +/R PU 2 dx dy; we also consider combinations of these properties. I t is known that
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