Related papers: Minimal surface doublings and electrostatics for S…

We prove that for any large enough $m \in\mathbb{N}$, the genus $\gamma=m+1$ equator-poles minimal surface doubling of the equatorial two-sphere $\Sigma^0 = \mathbb{S}^2_{\mathrm{eq}}$ in the round three-sphere $\mathbb{S}^3$, which has two…

We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as…

This paper investigates minimal $n$-dimensional submanifolds in the Euclidean space that are $(n-2)$-umbilic, meaning they carry an umbilical distribution of rank $n-2$. We establish a correspondence between the class of minimal…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity…

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some…

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of \pi/m. They are invariant…

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…

Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,\alpha}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,\alpha}$ two-valued…

This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to…

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry…

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge of the…

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected…

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max…

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations.…

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the…

In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical…