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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 of $l$, and a minimal surface which contains both $ a $ and the translated $l$.
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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 function 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 undergraduate analysis course for students of Mathematics at European universities. No prior knowledge of differential geometry is assumed.
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.
We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $mathbb D$ in $mathbb C$ into the unit ball $mathbb B^n$ in $mathbb R^n$, $nge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $nge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $mathbb Dto mathbb B^n$. This implies that conformal harmonic immersions $M to mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$mathrm{e}$ metric on $M$ and the Cayley-Klein metric on the ball $mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $mathbb D$ onto affine discs in $mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $mathbb R^n$ based on minimal surfaces.
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