arXiv research

New minimal discs and annuli found in ellipsoids.

Constructs minimal annuli with free boundary in hyperbolic 3-space.

New minimal annuli found in unit ball, solving old problems.

The critical catenoid is uniquely determined by certain symmetries of its boundary.

Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.

Constructs minimal surfaces near the boundary of a ball.

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.

We explicitly classify all $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the $k$-th normaliz…

Constructs minimal surfaces in a 3-ball using PDE gluing.

The study finds many Möbius bands and annuli on toroids.

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

The study constructs free boundary CMC annuli in spherical and hyperbolic balls.

Minimal surfaces reflect across spheres, proving annulus uniqueness.

Proves uniqueness of catenoid-like shapes in a ball.

New minimal surfaces found with spherical curvature lines.

Minimal surfaces span periodic curves in 3D space.

The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.

Strict convexity is essential for compact minimal surfaces in curved spaces.

Study proves all free boundary CMC annuli are of finite type.

Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.

We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply-connected. We show that if two components of the free boundary are close, then the solution locally resembles an entire solution discovered by Haus…

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Study calculates first $p$-widths of unit disk.

New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.

Free boundary minimal submanifolds with boundaries on concentric spheres

Minimal annuli constructed in PSL2 via variational method.

Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.

We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…

Study on surfaces minimizing elastic energy with boundary constraints.

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

New examples show non-rotational annuli in a ball, solving a uniqueness problem.

Study eigenvalues of magnetic Steklov problem on Riemannian annuli.

In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say that the ends are horizontal when they are graphs of $\mathcal{C}^{2, α}$ functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show that one can …

Study minimal annuli in a slab, estimating their area.

The paper explores geometric properties of free boundary hypersurfaces in balls.

Sharp lower bound found for area of vector fields on spherical annuli.

We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli hav…

In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic harmonic maps $G : \…

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the si…

Improved flatness in annuli using PDE methods.

A semigroup of annuli integrates a central extension of vector fields on S^1.

We study minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set…

The paper proves a statement about surfaces diffeomorphic to annuli.

We study the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$. These are singly periodic annuli with constant mean curvature $H>1/2$ in $\mathbb{H}^2\times\mathbb{R}$, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither verti…

Estimates the growth of Morse index for free boundary minimal hypersurfaces.

Proves smoothness of minimal surfaces near polyhedral boundaries.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

The study proves the existence of free boundary minimal disks in convex regions.