arXiv research

Generalizes Bestvina's $\mathcal{Z}$-boundaries to coarse $\mathcal{Z}$-boundaries.

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined…

Study on magnetic field and potential systems to prove rigidity results.

Characterizes relative hyperbolicity using Morse and contracting boundaries.

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $ρ_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper global uniqueness …

The study examines stationary surfaces with boundaries and their properties.

Compact hypersurfaces minimize area in convex cones with free boundary.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

Minimal crystallizations bound for 3-manifolds with boundary.

Let $\mathcal{M}$ be a moduli space of polystable rank 2-bundles bundles with fixed determinant (a moduli space of $\mathrm{PU}(2)$-instantons) on a Gauduchon surface with $p_g=0$ and $b_1=1$. We study the holomorphic structure of $\mathcal{M}$ around a circle $\mathcal{T}$ of regular reductions. Our model space is a "…

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regu…

In this paper we study the boundary at infinity of the curve complex $\mathcal{C}(S)$ of a surface $S$ of finite type and the relative Teichmüller space $\mathcal{T}_{el}(S)$ obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. $\mathcal{C}(S)$ an…

Researchers solve boundary and scattering rigidity problems for magnetic systems.

The study classifies geometrically finite polynomials on the boundary of Blaschke products.

The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.

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

New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.

New findings on lightconvex boundaries in Finslerian spacetimes.

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

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

Adapts viscosity method for free boundary minimal surfaces.

Efficient triangulations help in understanding 3-manifold boundaries.

This paper proves that one-relator groups have a weak Z-structure.

A $\mathcal{Z}$-structure on a group $G$ was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $\mathcal{EZ}$-structure. The…

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used we…

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut poin…

Symplectic coordinates found on projective structures on orbifolds.

We prove that given a $\mathcal{C^\infty}$ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to $\mathcal{C}^1$ manifolds, and that in dimensions $2,3$ and 4 it also holds for $PL$ manifolds. We employ the main res…

In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$. Let $g\in G$ and let $\mathcal{F}_g$ be the fixed-point set of $g$ in the boundary $\partial X$. Then we show that $\mathcal{F}_g=L(Z_g)$, where $Z_g$ is …

The paper investigates surfaces with boundary and mean curvature depending on the Gauss map.

New proof for hyperbolic groups using contracting boundaries of cusped spaces.

Algebras of smooth functions help reconstruct bulk topological types.

Let $(X_0,\mathcal{F}_0)$ be a compact manifold with boundary endowed with a foliation $\mathcal{F}_0$ which is assumed to be measured and transverse to the boundary. We denote by $Λ$ a holonomy invariant transverse measure on $(X_0,\mathcal{F}_0)$ and by $\mathcal{R}_0$ the equivalence relation of the foliation. Let…

New index formulae derived for operators on boundary groupoids.

Study proves Liouville theorem for specific curvature equations with boundary conditions.

We prove that if a smoothly bounded strongly pseudoconvex domain $D \subset \mathbb C^n$, $n \geq 2$, admits at least one Monge-Ampère exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $τ: \overline D \to [0,1]$, which is $\mathcal C^\infty$ at all points except possibly at the unique minimum poi…

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theo…

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space def…

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Se…

We show that if $\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\mathcal{L}$ are all in the interval $(-δ,δ)$ for a fixed $δ\in[0,1)$ and no complimentary region of $\mathcal{L}$ is an interval bundle over a surface, then each bo…

Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$ marked points on the boundary) with an arbitrary oriented surface with a number of la…

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular Kähler-Einstein metrics when ${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain a formula computing the volu…

Contractible diffeomorphism groups on lens spaces derived from Morse-Bott foliations.

Let $μ$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost every sample path of the random walk on $(\text{Out}(F_N),μ)$, when realized in Culle…

Stokes' theorem's boundary maximizes entropy.

Lower bounds for PL 4-manifolds with boundary are improved.

The paper studies diffeomorphisms of a specific foliation on a Klein bottle.