Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

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...

Half-space Theorem, Embedded Minimal Annuli and Minimal Graphs in the Heisenberg Group

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil3 is entire. Also, the sister surface of an entire minimal graph in Nil3 is an entire constant mean curv...

On Properly Embedded Minimal Surfaces with Three Ends

We classify all complete embedded minimal surfaces in R3 with three ends of genus g and at least 2g + 2 symmetries. The surfaces in this class are the Costa-HoffmanMeeks surfaces that have 4g + 4 symmetries in the case of a flat middle end. The proof consists of using the symmetry assumptions to deduce the possible Weierstrass data and then studying the period problems in all cases. To handle t...

Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M ⊂ R with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Pérez, and Ros [64] states that such an M must be congruent to a homothetic scaling of one of the classical examples foun...

Embedded Minimal Annuli in R Bounded by a Pair of Straight Lines