June 29, 2007 There has been a recent resurgence of interest in the subject of minimal surfaces in product three-manifolds, for example H ×R [Ros02], [MR04], [MIR05], [NR02], [Dan06], [FM05], [HET05]. The purpose of this note is to describe a construction and some examples originating in the harmonic maps literature whose consequences for minimal surfaces seem to have escaped notice. We organiz...

The main topic of this work is the definition and investigation of a nonlinear energy for maps with values in trees and graphs and the analysis of the corresponding nonlinear Dirichlet problem. The nonlinear energy is defined using a semigroup approach based on Markov kernels and the nonlinear Dirichlet problem is given as a minimizing problem of the nonlinear energy. Conditions for the existen...

(1) In Section 2, we define the notion of harmonic maps and quadratic differentials. Then we give a harmonic map proof of Teichmuller’s theorem by Wolf. (See [2], [9]) (2) In Section 3, we explain a compactification of Teichmuller spaces by Wolf using harmonic maps. (See [2], [4], [10], [11]) (3) In Section 4, we review the Morgan-Shalen compactification and the Korevaar-Schoen limit. Then we g...

In this paper, we present an efficient and robust technique for surface remeshing based on harmonic maps. We show how to ensure a one-to-one mapping for the discrete harmonic map and introduce a cubic representation of the geometry based on curved PN triangles. Topological and geometrical limitations of harmonic maps are also put to the fore and discussed. We show that, with the proposed approa...

We develop an efficient numerical method for analyzing second harmonic generation (SHG) in two-dimensional photonic crystals composed of nonlinear circular cylinders embedded in a linear background medium. Instead of solving the governing inhomogeneous Helmholtz equation for the second harmonic wave in the entire structure directly, we define and solve a locally generated second harmonic field ...