Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

Abstract

We prove that, given a compact Riemann surface $$\Sigma $$ and disjoint finite sets $$\varnothing \ne E\subset \Sigma $$ and $$\Lambda \subset \Sigma $$ , every map $$\Lambda \rightarrow \mathbb {R}^3$$ extends to a complete conformal minimal immersion $$\Sigma \setminus E\rightarrow \mathbb {R}^3$$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $$\mathbb {R}^3$$ with finite total curvature. To this respect we provide, for each integer $$r\ge 1$$ , a set $$A\subset \mathbb {R}^3$$ consisting of $$12r+3$$ points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface $$X:M\rightarrow \mathbb {R}^3$$ , then the absolute value of the total curvature of X is greater than $$4\pi r$$ . In order to prove this result we obtain an upper bound for the number of intersections of a complete immersed minimal surface of finite total curvature in $$\mathbb {R}^3$$ with a straight line not contained in it, in terms of the total curvature and the Euler characteristic of the surface.