arXiv research

New minimal discs and annuli found in ellipsoids.

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

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

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

Deep learning method solves American options with free boundary using Landau transformation.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.

Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.

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

The study finds many Möbius bands and annuli on toroids.

Proposes neural networks for solving complex free boundary problems.

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the correspondi…

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

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…

Foundations for free boundary Brakke flows established.

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…

For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $Σ_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $Σ_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $Σ_g$ converge as varifolds to the union of the d…

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $ρ$ = $2r/ σ^2$, where $r$ is the interest rate and $σ$ is the volatility. We use perturbation met…

Develops a method for conformal parameterization of point clouds without fixed boundaries.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.

We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.

Study proves minimality of certain hyperplane intersections in wide cones.

Constructs minimal surfaces near the boundary of a ball.

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.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …

Sharp geometric inequalities for free boundary hypersurfaces in balls.

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

New solutions found for a complex boundary problem.

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

Optimal inequality for free boundary hypersurfaces in convex domains.

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

Proves existence of mean curvature flow with surgery for free boundary surfaces.

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…

Study finds rigid properties of boundary-free hypersurfaces in specific data sets.

Study on stability of free boundary Willmore problem using new gradient inequality.

Connectivity proven in large rank Gromov boundary of free factor complex.

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

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

Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.

In this paper we study a utility maximization problem with both optimal control and optimal stopping in a finite time horizon. The value function can be characterized by a variational equation that involves a free boundary problem of a fully nonlinear partial differential equation. Using the dual control method, we der…

The paper classifies CMC free boundary hypersurfaces in rotational domains.

The paper proves the existence and properties of geodesics on convex surfaces.

Proves smoothness of minimal surfaces near polyhedral boundaries.

Straight lines are a basin of attraction for the elastic flow at least to level 1.9615π.

New spectral estimates for minimal surfaces with boundary conditions.

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…

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

Smooth solutions up to evolving free boundaries for degenerate equations.