arXiv research

We study the existence and uniqueness problem of compact minimal vertical graphs in $\mathbb{H}^n\times\mathbb{R}$, $n\geq 2$, over bounded domains in the slice $\mathbb{H}^n\times\{0\}$, with non-connected boundary having a finite number of $C^0$ hypersufaces homeomorphic to the sphere $\mathbb{S}^{n-1}$, with prescri…

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

Geodesic boundaries of surfaces with constant mean curvature are studied.

New method classifies $\mathbb C$-boundaries up to 6 crossings.

Study asymptotic behavior of translators in hyperbolic product space.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

Characterizes the Z-genus of boundary links using Blanchfield forms.

The study classifies quasi-Einstein manifolds with boundary.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichmüller …

Classifies $4$-manifolds with specific properties and fundamental group.

Simple closed curves in ε-boundaries separate sets in the plane.

The study finds disks for certain constant mean curvature surfaces in a specific 3D space.

The study finds minimal surfaces in complex space forms are often totally geodesic.

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

Harmonic functions $u:{\mathbb R}^n \to {\mathbb R}^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,{\mathcal I},ω)$. To this system one associates the space of conservation laws ${\mathcal C}$. They provide necessary conditions for $g:{\mathbb S}^{n-1} \to M$ …

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable general…

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

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…

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

The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}$ (\cite{Pacard}). The exist…

Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2\times\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.

Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same boundary as $D$. Then $I(D)= λ_P^n \cdot Area(D)$. If $P$ is not complex, $λ_P^n < \frac{3π}{…

Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.

Study the topology of Milnor boundaries for real analytic map germs.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $Π$ is Fredholm by showing that the stationary vacuum equations (combined with p…

The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected $…

Locally connected boundaries proven for relatively hyperbolic groups.

Let Ωand \tildeΩ be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: Ω\to \tildeΩ such that the graph Σ= \{(x,f(x)): x \in Ω\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and…

Irreducible groups cannot be free if they ergodically act on a boundary.

Unified rigidity theorem for Plateau surfaces in $\mathbb{B}^n$.

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

In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

Study resolves flow through cylindrical singularities, proving nonfattening.

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is eve…

Minimal surfaces in spheres constructed from symmetry reductions of ODEs.

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized …

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

Study of disks in complex projective space with specific fundamental groups.

New manifold construction yields Baire-1 functions as cohomotopy groups.

The study proves Gromov hyperbolicity for certain complex domains.

Paper proves translating solutions for a specific flow in a product manifold.

Study extended Bogomolny equations on curved space with special boundary conditions.

This paper embeds surfaces in 3D spheres and balls with minimal area.