Related papers: Minimal surfaces in $S^3$ foliated by circles

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…

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5 space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space.…

Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only…

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of…

We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal)…

We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that…

In R^3, let M be the infinite union of unit spheres whose centers lie at even integers on the x-axis; every pair of consecutive spheres touches at (2m+1, 0, 0). Desingularizing these point contacts yields Delaunay's classical constant mean…

We study minimal transverse foliations which are $R$-covered. If in addition the dimension of the ambient manifold is $3$, and the foliations are Anosov foliations we give necessary and sufficient conditions for the intersected foliation to…

We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of…

Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. K\"ahler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we…

We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric $Q_{2}\cong \mathbb S^{2}\times \mathbb S^{2}$, formulated via a flat family of connections $\{\nabla^\lambda\}_{\lambda\in \mathbb…

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher…

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general…

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…

It is shown that analytic conformal submersions of $S^3$ are given by intersections of (not necessary closed) complex surfaces with a quadratic real hyper-surface in $\mathbb{C}P^3.$ A new description of the space of circles in the 3-sphere…

Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results…

In this work we present a new family of properly embedded free boundary minimal hypersurfaces of revolution with circular boundaries in the horizon of the $n$-dimensional Schwarzschild space, $n\geq3$. In particular, we answer a question…

It is well known that the only surfaces that are simultaneously minimal in $\mathbb{R}^3$ and maximal in $\mathbb{L}^3$ are open pieces of helicoids (in the region in which they are spacelike) and of spacelike planes (O. Kobayashi 1983).…

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space $\mathbb{E}^3$ is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same…

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean $(n+1)-$space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized…