Abstract:
This thesis is concerned with motivating and contributing to the development of pluripotential theory on algebraic curves. The first two sections comprise a self-contained exposition of the necessary background information required to understand pluripotential theory on algebraic curves. The following two sections discuss some of the problems that arise in pluripotential theory when on algebraic curves. In particular, we simplify some of the proofs from Ma'u's paper [10], study directional Chebyshev constants under transformation by polynomial maps between algebraic curves and extend Ma'u's work to general algebraic curves. We also study a possible generalisation of the Robin's function to algebraic curves and in doing so prove an analogue of Sadullaev's theorem on Green's functions ([16], Prop. 3.4) for the Robin's function of a compact set KcC².