arXiv research

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …

Curvature bounds preserved in length-minimizing disks.

New minimal discs and annuli found in ellipsoids.

In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free bound…

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying c…

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

We construct and analyze minimal disc stackings with bounds on their Morse index.

Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadra…

In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…

Constructs minimal surfaces near the boundary of a ball.

Minimal discs count as knot invariants in hyperbolic 4-space.

Sharp estimate on harmonic maps at conformal points in balls.

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conf…

Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.

Study on surfaces minimizing elastic energy with boundary constraints.

Metric spaces with upper curvature bounds have controlled Dehn functions.

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…

In 1996, Nadirashvili used Runge's theorem to produce a complete minimal disc inside a ball in R^3. In this paper we generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic c…

Let $F_g$ be a closed orientable surface of genus $g$. A set $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if $F_g\setminus Ω$ is a union of $b$ topological discs for some $b\geq 1$. A filling system is called \em…

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnor's…

A fast metric learning framework using Gershgorin disc alignment.

The study proves a geometric inequality for surfaces with genus G.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem…

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition o…

A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs.

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surfa…

New knots found with tough, unsliceable discs.

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization…

Classifies ancient flows in a disc with boundary.

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

Let $Γ$ be either the infinite cyclic group $\mathbb{Z}$ or the Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has fundamental group $Γ$. We classify the $Γ$-homotopy ribbon slice discs for $K$ up to topological ambien…

The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.

Study smooth manifolds using disc-presheaves.

This note characterizes monohedral tilings of regular polygons with up to three tiles.

Study horizontal discs in fat distributions, proving their existence.

Study investigates minimal surfaces in spherical caps, extending previous findings.

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

The paper classifies homotopy ribbon discs with specific groups.