The approach applies to complex constructs, for example arbitrary geometric embeddings of graphs rather than surface meshes, lattice subsets, and meshes on higher genus surfaces than spheres.
The greater topological complexity of arbitrary graph embeddings and meshes on higher genus surfaces can result in scaffolding strand routes that are knotted in 3 space, so we also present necessary caveats for these settings.
Seven examples including open, semiclosed, closed, zero-genus, high-genus surfaces and real-world scanned objects, described in free-form, parametric and implicit forms illustrate the good performance of our approach and its superiority over previous approaches in terms of accuracy and generality.
We next mention a result concerning the Willmore energy of surfaces of high genus: (Kuwert et al. [35]) There exists a sequence of real numbers (beta _g in (4pi,8pi )) such that (lim _{g rightarrow infty } beta _g = 8pi ) and ({mathscr{W}}(Sigma ) ge beta _g) for every immersed surface (Sigma ) in (S^3) of genus (g).
Wen, X. G. & Niu, Q. Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces.
In this work we show the existence of higher genus minimal capillary surfaces by a perturbation method.
Finally, based on density functional theory calculations, the electronic properties of the high genus I-WP were examined for the first time finding semiconducting, semimetallic, and metallic behaviors.
A similar result, whose statement requires the concept of Riemann surface, is true for surfaces of higher genus with finitely many boundary components.
This changed dramatically in the late 1960s, when Lawson discovered an infinite family of embedded minimal surfaces of higher genus: (Lawson [37]) Given any pair of positive integers (m) and (k), there exists an embedded minimal surface (Sigma ) in (S^3) of genus (mk).
Interestingly, the corresponding colouring problem concerning the number of colours required to colour maps on surfaces of higher genus was completely solved a few years earlier; for example, maps on a torus may require as many as seven colours.
