Ildefonso Castro-Infantes: Approximation and interpolation by conformal minimal immersions of finite total curvature and optimal hitting problem
Vir: Seminar za kompleksno analizo
Title: Approximation and interpolation by conformal minimal immersions of finite total curvature and optimal hitting problem.
Abstract: Let M be a Riemann surface. A conformal minimal immersion X: M\to\R^3 is said to be of total finite curvature if TC(X):= \int_M Kds^2=-\int_M |K|ds^2>-\infty, where K denotes the Gauss curvature of the conformal minimal immersion X. Lopez has proved that a Mergelyan approximation theorem holds for the family of conformal minimal immersions of finite total curvature. In this talk we prove an interpolation theorem for this family of minimal surfaces. That is, given a compact Riemann surface \Sigma and disjoint finite sets \emptyset\neq E\subset\Sigma and \Lambda\subset\Sigma, every map \Lambda \to R^3 extends to a complete conformal minimal immersion \Sigma\setminus E \to R^3 with finite total curvature. We obtain this result as a consequence of a more precise one providing approximation, interpolation of given finite order, and control on the flux. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in R^3 with finite total curvature. To this respect we provide, for each integer r\ge 1, a set A \subset 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 \to R^3, then the absolute value of the total curvature of X is greater than 4\pi r.
Seminar bo v predavalnici 3.06 na Jadranski 21. Vljudno vabljeni!
Josip Globevnik in Franc Forstnerič