The first Steklov eigenvalue, conformal geometry, and minimal surfaces

The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure. Mathematics Subject Classification (2000)....

Conformal Maps and Minimal Surfaces

On the First Eigenvalue of a Fourth Order Steklov Problem

We prove some results about the first Steklov eigenvalue d1 of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality [9] may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d1 for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problem...

The topology, geometry and conformal structure of properly embedded minimal surfaces

Let M denote the set of connected properly embedded minimal surfaces in R with at least two ends. At the beginning of the past decade, there were two outstanding conjectures on the asymptotic geometry of the ends of an M ∈ M that were known to lead to topological restrictions on M . The first of these conjectures, the generalized Nitsche conjecture, stated that an annular end of such a M ∈ M is...

The Geometry and Conformal Structure of Properly Embedded Minimal Surfaces of Finite Topology in R

In this paper we study the conformal structure and the asymptotic behavior of properly embedded minimal surfaces of finite topology in . One consequence of our study is that when such a surface has at least two ends, then it has finite conformal type, i.e., it is conformally diffeomorphic to a compact Riemann surface punctured in a finite number of points. Except for the helicoid and a recently...