Extremal length and Dirichlet problem on Klein surfaces

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in Rn such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in Rn can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fi...

Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

Let T be a triangulation of a Riemann surface. We show that the 1-skeleton of T may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from T by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the cl...

Miranda-persson’s Problem on Extremal Elliptic K3 Surfaces

In one of their early works, Miranda and Persson have classified all possible configurations of singular fibers for semistable extremal elliptic fibrations on K3 surfaces. They also obtained the Mordell-Weil groups in terms of the singular fibers except for 17 cases where the determination and the uniqueness of the groups were not settled. In this paper, we settle these problems completely. We ...

Extremal Length and Removable Boundaries of Riemann Surfaces

On Extremal Elliptic K3 Surfaces

In [MP2], R.Miranda and U.Persson have classified possible semi-stable fibrations. The determination of all semi-stable fibrations has been done in [MP3] and [ATZ]. In this paper, we first classify all possible configurations of unsemi-stable fibrations (cf. Theorem 2.4). Then we calculate the possible Mordell-Weil Groups for Case(A) (cf.Theorem 3.1), i.e., the case where each singular fibre of...