arXiv research

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

Embedded surfaces in a ball have any genus and connected boundary.

Clusters on simple manifolds have connected boundaries.

Study on connectivity of Morse boundaries of Coxeter groups.

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…

In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.

The boundary of hyperbolic groups is locally simply connected.

Locally connected boundaries of Coxeter groups are studied.

The paper proves hyperbolic groups are semistable and their boundaries are linearly connected.

Spheres in curve graphs are connected, proving Gromov boundary linearity.

Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold wit…

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.

Projective structures are mostly rigid at the boundary but some are not.

Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of …

Study of Dirac operator with chiral boundary conditions on spin manifolds.

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.

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

This work connects the Hessian to the decision boundary complexity in neural networks.

Connected components of Morse boundaries are studied in graph of groups.

A seminal result in geometric group theory is that a 1-ended hyperbolic group has a locally connected visual boundary. As a consequence, a 1-ended hyperbolic group also has a path connected visual boundary. In this paper, we study when this phenomenon occurs for CAT(0) groups. We show if a 1-ended CAT(0) group with iso…

Compact manifolds with boundary have finite type mapping class groups.

New characterizations for manifolds with boundary rigidity results.

Proves well-posedness for Einstein equations with totally geodesic timelike boundary condition.

In this paper, we shall prove that any Heegaard splitting of a $\partial$-reducible 3-manifold $M$, say $M=W\cup V$, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of $n$ manifolds $M_{1},..., M_{n}$ where $M_{i}$ is either a solid torus or a $…

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with conne…

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the…

For a given family of smooth closed curves $γ^1,...,γ^α\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^α$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimi…

Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.

The paper studies the connectedness of a graph's boundary for surfaces.

We provide some language for algebraic study of the mapping class groups for surfaces with non-connected boundary. As applications, we generalize our previous results on Dehn twists to any compact connected oriented surfaces with non-empty boundary. Moreover we embed the `smallest' Torelli group in the sense of Putman …

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

Connected boundaries of strata of differentials are always connected in various compactifications.

The paper shows connections can be uniquely determined by their boundary data.

We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1$, we prove that the scattering matrix …

We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

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…

Let $(X_1, \bar g_1)$ and $(X_2, \bar g_2)$ be two compact Riemannian manifolds with boundary $(M_1,g_1)$ and $(M_2,g_2)$ respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundar…

The study proves unique static manifolds with positive scalar curvature and boundary.

Computes mapping class groups of 4-manifolds with boundary.

Essential triangulations connect via specific moves in 3-manifolds.

Study compact PL 4-manifolds with special handle decompositions.

Improved method for numerical conformal mappings on complex domains.

Study shows saddle connection graph's geometry and quasi-isometry properties.

We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is th…

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 $…

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…