Maximum principles for minimal surfaces in \(\mathbb{R}^ 3\) having noncompact boundary and a uniqueness theorem for the helicoid
For the sake of simplicity two main results will be quoted: ``Let \(M_1\) be a complete and \(M_2\) a properly immersed minimal surface in \(\mathbb{R}^3\) such that the principal curvatures of \(M_1\) and \(M_2\) are bounded in modulus by some constant \(k\). We assume furthermore \(M_1\cap M_2= \emptyset\), \(\min\{d(M_1, \partial M_2), d(M_2, \partial M_1)\}\geq \delta> 0\), and that \(M_1\) is ``bandlike'', i.e., the radius of geodesic discs contained in the interior of \(M_1\) is bounded by some fixed constant \(L\). Then it follows that \(d(M_1, M_2)> 0\)'' (where \(d(A, B)= \inf\{|p- q|\mid p\in A, q\in B\}\), \(A, B\subset \mathbb{R}^3\)). This theorem is derived from the Harnack inequality for elliptic differential equations. The second result, which is an application of the first one, reads as follows: ``Let \(S\) be a line segment through the origin in the \((x, y)\)-plane with endpoints \(p_1\) and \(p_2\) such that \(\max\{|p_1|, |p_2|\}\leq \varepsilon_0\). Let \(M\) be a complete properly immersed minimal surface, with bounded curvature and with boundary \(h_1\cup h_2\), where \(h_1\) and \(h_2\) are the helices generated by \(p_1\) and \(p_2\), and such that \[ \lim_{(x, y, z)\to \infty} {x^2+ y^2\over z^2}= 0,\qquad (x, y, z)\in M. \] Then \(M\) coincides with the helicoidal domain generated by \(S\).'' Here \(\varepsilon_0= \text{sinh}(u_0)= 1.50888\dots\) is defined by \(\text{cosh}(u_0)= u_0\text{sinh}(u_0)\).
scientific article; zbMATH DE number 824169
scientific article