Complete Minimal Surfaces and the Puncture Number Problem

Existence of Algebraic Minimal Surfaces for an Arbitrary Puncture Set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

The Minimal Logically-Defined NP-Complete Problem

The Cauchy problem for Lie-minimal surfaces

In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

Hyperbolic Complete Minimal Surfaces with Arbitrary Topology

We show a method to construct orientable minimal surfaces in R3 with arbitrary topology. This procedure gives complete examples of two different kinds: surfaces whose Gauss map omits four points of the sphere and surfaces with a bounded coordinate function. We also apply these ideas to construct stable minimal surfaces with high topology which are incomplete or complete with boundary.

Complete Properly Embedded Minimal Surfaces in R3

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface 2 ⊂ R3 with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature. In this short paper, we apply estimates and ideas from [CM4] to study the en...