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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.
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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 $uin H^{1/2}(partial S,mathcal N)$. These maps are critical points of the nonlocal energy begin{equation}E(f;g):=int_Sbig| ablawidetilde ubig|^2,dtext{vol}_g,end{equation} where $widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $frac 12$-energies at each boundary component of $partial S$ (suitably identified with the circle $mathcal S^1$), plus a quadratic term which is continuous in the $H^s(mathcal S^1)$ topology, for any $sinmathbb R$. We show the $C^{k-1,delta}$ regularity of $frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmuller space of $S$.
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 $pi_n(mathcal{N})={0}$ then there exists a minimizing $W^{s,frac ns}$-harmonic map homotopic to $u$. If $pi_n(mathcal{N}) eq {0}$, then we prove that there exists a $W^{s,frac{n}{s}}$-harmonic map from $mathbb{S}^n$ to $mathcal{N}$ in a generating set of $pi_{n}(mathcal{N})$. Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when $frac{n}{s} eq 2$ one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing $W^{s,frac{n}{s}}$-maps into manifolds.
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