In this article, we interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $mathbb{L}^3$ to another real analytic spacelike curve $c$, which is close enough to $a$ in a certain sense, by a maximal surface using inverse funct
ion theorem for Banach spaces. Using the same method we also interpolate a given real analytic curve $a$ in Euclidean space $mathbb{E}^3$ to another real analytic curve $c$, which is close enough to $a$ in a certain sense, by a minimal surface. The Bjorling problem and Schwartzs solution to it play an important role.
In this article we present an elementary introduction to the theory of minimal surfaces in Euclidean spaces $mathbb R^n$ for $nge 3$ by using only elementary calculus of functions of several variables at the level of a typical second-year undergradua
te analysis course for students of Mathematics at European universities. No prior knowledge of differential geometry is assumed.
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 consider immersions of a Riemann surface into a manifold with $G_2$-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.
Let $a: Ito mathbb{R}^3 $ be a real analytic curve satisfying some conditions. In this article, we show that for any real analytic curve $l:Ito mathbb R^3$ close to $a$ (in a sense which is precisely defined in the paper) there exists a translation o
f $l$, and a minimal surface which contains both $ a $ and the translated $l$.