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G. A. Lobos

Publications and source records attributed to G. A. Lobos.

3 recordsLinked to original sources
arXiv2025

The Green's Function on Rhombic Flat Tori

We obtain the Green's function $G$ for any flat rhombic torus $T$, always with numerical values of significant digits up to the fourth decimal place (noting that $G$ is unique for $|T|=1$ and $\int_TGdA=0$). This precision is guaranteed by the strategies we adopt, which include theorems such as the Legendre Relation, properties of the Weierstraß\,P-Function, and also the algorithmic control of numerical errors. Our code uses complex integration routines developed by H. Karcher, who also introduced the symmetric P-Weierstraß\,function, and these resources simplify the computation of elliptic functions considerably.

math.NA↗
arXiv2013

Explicit minimal Scherk saddle towers of arbitrary even genera in $\R^3$

Starting from works by Scherk (1835) and by Enneper-Weierstraß\ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}). However, Traizet's construction is implicit and excludes {\it towers}, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martin and Ramos Batista (see \cite{Martin}), all of them with genus one. For genus two, the first such towers were constructed in 2010 (see \cite{Valerio2}). Back to 2009, implicit towers of arbitrary genera were found in \cite{HMM}. In our present work we obtain {\it explicit} minimal Scherk saddle towers, for any given genus $2k$, $k\ge3$.

math.DG↗