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In this paper, we discuss a one parameter family of complex Born-Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B-I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We find that the action or the energy of this family of solitons remains invariant in this family and find that the well-known Lorentz symmetry of the B-I equations is responsible for it.
Using Ramanujans identities and the Weierstrass-Enneper representation of minimal surfaces and the analogue for Born-Infeld solitons, we derive further non-trivial identities.
We discuss quantum analogues of minimal surfaces in Euclidean spaces and tori.
Given a $C^k$-smooth closed embedded manifold $mathcal Nsubset{mathbb R}^m$, with $kge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $partial S eqemptyset$, we consider $frac 12$-harmonic maps $uin H^{1/2}(partial S,mathcal N)$.
Let $Sigma$ a closed $n$-dimensional manifold, $mathcal{N} subset mathbb{R}^M$ be a closed manifold, and $u in W^{s,frac ns}(Sigma,mathcal{N})$ for $sin(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if $pi_n(mathcal{N})=