arXiv research

Survey of non-uniqueness in anisotropic Calderón problem with disjoint boundary data.

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{…

Uniform bounds found for Sierpinski carpet hyperbolic components.

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $Γ_1$ and $Γ_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator $Λ_{Γ_1,Γ_2}$. This operator corresponds the boundary measure…

Maximizes filling systems on surfaces with given boundary components.

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüll…

Study calculates index for non-compact manifolds with boundary.

We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping …

A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In this paper, we give various results about such collections of curves and the assoc…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the…

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not contradictory result that although one can sometimes embed arbitrarily many surfaces in…

Infinite clique of rays in plane minus Cantor set.

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show that there exist two such surfaces if and only if M is neither a Z_2 homology sol…

Formula calculates surface torsion via pants decompositions.

For two nearby disjoint coassociative submanifolds C and C' in a G_2-manifold, we construct thin instantons with boundaries lying on C and C' from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

Consider a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary is the disjoint union of two pieces: the horizon boundary and the outer boundary, where the horizon boundary consists of the unique closed minimal surfaces in the manifold and the …

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…

BDSG generates samples on distribution boundaries, improving anomaly detection.

We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion c…

We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$…

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capill…

Study the boundaries of ε-neighborhoods of planar sets, showing their structure and curvature.

A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-ma…

Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary of S perpendicularly are coordinates on the Teichmueller space T(S). We compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and we prove that i…

We construct a relation among right-handed Dehn twists in the mapping class group of a compact oriented surface of genus g with 4g+4 boundary components. This relation gives an explicit topological description of 4g+4 disjoint (-1)-sections of a hyperelliptic Lefschetz fibration of genus g on the manifold {CP}^2#(4g+5)…

Haken n-manifolds are aspherical manifolds, defined and studied by B. Foozwell and H. Rubinstein, that can be successively cut open along essential codimension-one submanifolds until a disjoint union of n-cells is obtained. Such manifolds come equipped with a boundary pattern, a particular kind of decomposition of the …

Decision trees are a popular technique in statistical data classification. They recursively partition the feature space into disjoint sub-regions until each sub-region becomes homogeneous with respect to a particular class. The basic Classification and Regression Tree (CART) algorithm partitions the feature space using…

In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singu…

Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish tha…

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k…

Reciprocity laws for line bundles on circle fibrations over complex manifolds.

We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disj…

Maps approximable by homeomorphisms under certain conditions.

A new spectrum recovers cobordism cut and paste groups of manifolds with boundary.

This paper classifies knots with simple curves in genus 2 handlebodies.

Study classifies static potentials on 3-manifolds, proving one-dimensionality under specific conditions.

A new classification method using disjoint centroids and normalized distance.

Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a, f^m(a))$ fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ …

Study area-minimizing currents with specific boundary properties.

Upper bounds on renormalized volume for Schottky groups derived from extremal lengths.

Study finds unique critical points for anisotropic surface energy.

TailGAN uses GANs to detect anomalies near data distribution tails.

Paper proposes a clustering algorithm for nonnegative data.

The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.

Study the APS index on manifolds with non-compact boundaries.

In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the bou…

We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form, namely, the type D structure may be described graphically by a disjoint union of loops. We develop a calculus for studying bordered invariants of this form and, in particular, provide a complet…