Related papers: Minimal surfaces in the three dimensional sphere w…

We prove the existence of complete minimal surfaces in $\mathbb{R}^3$ of arbitrary genus $p\, \ge\, 1$ and least total absolute curvature with precisely two ends -- one catenoidal and one Enneper-type -- thereby solving, affirmatively, a…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to…

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…

Let $\mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $\mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $\mathbb{S}_h$. We give a new…

Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry. Whilst minimal surfaces are well understood, Einstein manifolds remain far less so. This exposition synthesises together a set of…

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to…

The Clifford torus is a product surface in $\mathbb S^3$ and it is helicoidal. It will be shown that more minimal submanifolds of $\mathbb S^n$ have these properties.

In this thesis, we present various contributions to the study of free boundary minimal surfaces. After introducing some basic tools and discussing some delicate aspects related to the definition of Morse index when allowing for a contact…

This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $\lambda_1$-minimal immersion), which also play important roles in…

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously…

It is known that we can always 3-triangulate (i.e. divide into tetrahedra) convex polyhedra but not always non-convex ones. Polyhedra topologically equivalent to sphere with $p$ handles, shortly $p$-toroids, could not be convex. So, it is…

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let $g$ be a metric on the three-sphere $S^3$ satisfying $Ric(g) \geq 2 g$. If the volume of $(S^3, g)$ is no less than…

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the…

How can we visualize all the surfaces that can be made from the faces of the tesseract? In recent work, Aveni, Govc, and Rold\'an showed that the torus and the sphere are the only closed surfaces that can be realized by a subset of…

For each large enough $m\in\mathbb{N}$ we construct by PDE gluing methods a closed embedded smooth minimal hypersurface ${\breve{M}_m}$ doubling the equatorial three-sphere $\mathbb{S}_{\mathrm{eq}}^3$ in $\mathbb{S}^4(1)$, with…

We construct many closed, embedded mean curvature self-shrinking surfaces $\Sigma_g^2\subseteq\mathbb{R}^3$ of high genus $g=2k$, $k\in \mathbb{N}$. Each of these shrinking solitons has isometry group equal to the dihedral group on $2g$…

This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under…

The goal of this paper is to establish the classification of all homogeneous surfaces of 3-sphere by using the moving frame method. We will show that such surfaces are 2-spheres and flat torus.

We prove that a minimal hypersurfaces $f\colon M^{3} \to \mathbb{Q}^4(c)$ with nonzero three distinct principal curvature cannot be isometrically immersed in $\mathbb{Q}^4(\tilde{c}), \ \tilde{c}\neq c$. In the other cases, we present a…

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary components. We show that the width for the area functional can be achieved by a bubble tree limit…