arXiv research

Sharp upper bound found for stable minimal surfaces.

Characterizes stable minimal capillary surfaces with specific angles.

Proof that stable minimal surfaces in 3D are flat.

Paper proves stable minimal surfaces in 3D are flat.

Paper studies curvature of stable surfaces meeting at a common boundary.

The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.

Recent work on stable minimal hypersurface singularities.

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable comp…

New minimal surfaces grow area very quickly.

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…

Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.

Flat minimal hypersurfaces in 4D space are always flat.

Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.

New stable minimal surfaces generalize classical Henneberg surface.

The study examines stable minimal surfaces in higher dimensions and provides bounds on their properties.

Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.

The paper constructs infinitely many surfaces with specific mean curvature.

The paper proves conditions for minimal surfaces to be holomorphic and stable.

It is shown that if a minimal ruled surface admits a Kähler Yamabe minimizer, then this metric must be generalized Kähler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.

Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic co…

Study stability of surfaces in spacetimes, proving new estimates and theorems.

The study examines stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

We discuss recent results on minimal surfaces and mean curvature flow, focusing on the classification and structure of embedded minimal surfaces and the stable singularities of mean curvature flow. This article is dedicated to Rick Schoen.

The paper extends a theorem about stable minimal surfaces to higher codimensions.

Minimal surfaces' area bounds proven equivalent, extending known results.

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space H^2xR; where H^2 is the hyperbolic plane. Under some geometric conditions on the asymptotic boundary of an oriented stable minimal surface immersed in H^2xR, it has infinite total curvature. In particular, we infe…

In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this pap…

New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.

The study establishes curvature estimates and convexity for a specific type of minimal surfaces.

Flat minimal hypersurfaces found in wedge-shaped domains.

Sharp bounds for charged Hawking mass in electrostatic space-times.

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depend…

Proves rigidity of stable minimal hypersurfaces in low dimensions.

Decomposes J-energy into simpler intersection numbers for stability analysis.

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian $3$-manifolds. We also produce new exampl…

We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constan…

The study of stable and index compact minimal submanifolds in Berger spheres.

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that $\bullet$ $\mathcal{C}_a$ is an unstable minimal surface with index o…

Classifies area-minimizing surfaces in R^4 as algebraic.

We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in d…

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable oriented double cover of a non-orientable minimal surface with a vertical handle atta…

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that $f$ is …

The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.

We show that the cone over a fibered face of a compact fibered hyperbolic 3-manifold is dual to the cone generated by the homology classes of finitely many curves called minimal stable loops living in the associated veering triangulation. We also present a new, more hands-on proof of Mosher's Transverse Surface Theorem…

New method improves curvature estimates for stable surfaces.