In the early 1980s, ST Yau conjectured that any compact Riemannian three-manifold admits
an infinite number of closed immersed minimal surfaces. We use min–max theory for the …

We present here a survey of recent spectacular successes in classical minimal surface
theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid …

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in
Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the …

In these notes we present some aspects of the basic theory on the geometry of a three-
dimensional simply-connected Lie group X endowed with a left invariant metric. This …

The aim of this paper is to analyze some of the relationships between oscillation theory for
linear ordinary differential equations on the real line (shortly, ODE) and the geometry of …

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts
approach. The Allen-Cahn approximation has proved to be a powerful alternative, and …

Meeks and Pérez extend their 2011 survey article" The classical theory of Minimal surfaces"
in the Bulletin of the American Mathematical Society to include other recent research results …

We develop a global theory for complete hypersurfaces in $\mathbb {R}^{n+ 1} $ whose
mean curvature is given as a prescribed function of its Gauss map. This theory extends the …

We consider a Riemannian submersion from a 3-manifold $\mathbb {E} $ to a surface $ M $,
both connected and orientable, whose fibers are the integral curves of a Killing vector field …

This is a survey on the global theory of constant mean curvature surfaces in Riemannian
homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston …