Related papers: Minimal surfaces - variational theory and applicat…
We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal…
Equations are derived for the shape of a hypersurface in $\mathbb{R}^N$ for which a rigid motion yields a minimal surface in $\mathbb{R}^{N+1}$. Some elementary, but unconventional, aspects of the classical case $N=2$ (solved by H.F. Scherk…
We give an overview of results on irregular complex surfaces of general type, discussing in particular the distribution of the numerical invariants self-intersection of a canonical divisor and holomorphic Euler characteristic for the…
We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential…
A class of spiral minimal surfaces in E^3 is constructed using a symmetry reduction. The new surfaces are invariant with respect to the composition of rotation and dilatation. The solutions are obtained in closed form %through the Legendre…
We construct examples of algebraic surfaces with interesting fundamental groups.
We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under M\"{o}bius transformations.…
Gradient, chemically modified, flat surfaces enable directed transport of droplets. Calculation of apparent contact angles inherent for gradient surfaces is challenging even for atomically flat ones. Wetting of gradient, flat solid surfaces…
We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of…
We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…
This Master's thesis presents a study of the basic properties of the s-fractional perimeter and of the regularity theory of the corresponding s-minimal sets. In particular, we give full detailed proofs for all the Theorems contained in the…
We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of…
A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can…
This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound…
This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an…
One of the main problems of the theory of dynamical systems is the determination of the existence of periodic orbits of a self-map and more generally, the structure of the set of periods. Define the minimum period of a class os self-maps of…
First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th…
This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and…
In this paper, we propose a variational formulation to study the singular evolution equations that govern the dynamics of surface modulations on crystals below the roughening temperature. The basic idea of the formulation is to expand the…
We study sheaves of differential forms and their cohomology in the h-topology. This allows to extend standard results from the case of smooth varieties to the general case. As a first application we explain the case of singularities arising…