arXiv research

Minimal surfaces' boundary points are always smooth.

The paper studies free boundary minimal surfaces with many boundaries and their convergence to closed minimal surfaces.

Proves smoothness of minimal surfaces near polyhedral boundaries.

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

New minimal surfaces found in ball with boundary constraints.

Study improves boundary smoothness for area-minimizing currents with complex boundaries.

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

New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.

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

New minimal discs and annuli found in ellipsoids.

The study bounds the topology of free boundary minimal surfaces in 3D manifolds.

Minimal graph level sets are concave if boundary is concave.

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

Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.

Study of free boundary minimal Möbius bands in spherical caps.

In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.

Researchers found multiple surfaces with same topological and symmetry properties.

In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of const…

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

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

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

We prove surfaces are unknotted with specific properties.

Local minimality proven for stable free-boundary minimal hypersurfaces.

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…

Paper studies minimal surfaces in curved spaces, proving existence and properties.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

Study on unique minimal hypersurfaces in rotational domains.

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 this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is…

New theorem on 3-manifolds with curvature and convex boundary.

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the …

Paper examines stability of minimizing metrics on manifolds with boundary.

Two minimal hypersurfaces in a ball intersect in any half-ball.

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

Study finds new minimal surfaces in Schwarzschild space.

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.

We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.

Constructs minimal surfaces near the boundary of a ball.

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

Clusters on simple manifolds have connected boundaries.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

Minimal surfaces reflect across spheres, proving annulus uniqueness.

Unique minimal surfaces near quadratic cones are identified.

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

A 3D area-minimizing current in R^5 has a 2-fold essential singularity.