GEOMETRIC INTEGRATION ON FRACTAL CURVES IN THE PLANE

We provide a method of integrating 1-forms over curves with zero area which need not have any tangent vectors, e.g. fractal curves. For Lipschitz curves this integral agrees with the usual line integral of 1-forms. The minimal requirement is that the box dimension of the curve should be smaller than one plus the Holder exponent satisfied by the form; when this condition is violated there are classical obstructions. We study the continuity of the integral by introducing a new norm (the d-flat norm) on the space of polyhedral 1-chains, well-adapted to Holder forms. We employ the abstract framework of Whitney's flat chains and cochains in this new context to obtain a sharper theory that reveals more clearly the nature of the obstruction Whitney discovered in the 30's.