arXiv research

Proposes cone embedding for better graph hierarchical structure representation.

Analytic sets with unique infinite tangent cone are algebraic.

The paper extends isometric embedding results to null cones and spheres.

In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.

Smooths out complex shapes into simpler forms.

Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle …

This paper proves area-minimizing cones over Grassmannian manifolds.

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-li…

The Kähler cone of a compact manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We write down the curvature tensor of this metric by embedding the Kähler cone in the space of hermitian metrics on the underlying manifold. After discussing weak functorality and complet…

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these y…

In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface $Γ\subset S^n$. The theory of isoparamet…

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete pro…

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-regular and mean convex (but not area-minimizing…

Holomorphic tensors on algebraic cones are invariant under certain group actions.

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…

If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $μ$ is an absolutely continuous measure on S, then we compute the tangent cone of P(M) at $μ$.

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or e…

We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.

New distances for comparing multivariate normal distributions.

We give a numerical characterization of the Kahler cone of a possibly singular compact analytic variety which is embedded in a smooth ambient space.

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform r…

Constructs examples of domains divided by groups in dimensions 3 and above.

Every nonflat conformal minimal surface is homotopic to a proper one.

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is…

Proofs show embedding conditions for complex joins and factors.

Minimal surfaces in spheres constructed from symmetry reductions of ODEs.

The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median su…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we de…

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_{i}$ of pointed hyperbolic cone-manifolds with topological type $(M,Σ)$, where $M$ is a closed, orientab…

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angl…

Smoothly approximates embeddings in Lorentzian manifolds.

The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach…

Paper equates torsions on wedge singularities.

In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.

Constructs simplified or complexified simplicial complexes.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency $2^{\aleph_0}$ at every point. It…

The nonlocal $s$-fractional minimal surface equation for $Σ= \partial E$ where $E$ is an open set in $R^N$ is given by $H_Σ^ s (p) := \int_{R^N} \frac {χ_E(x) - χ_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in Σ.$ Here $0<s<1$, $χ$ designates characteristic function, and the integral is understood i…

Defines smoothness of definable sets in o-minimal structures.

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant ρ_T, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.

Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.

Given a smooth, symmetric, homogeneous of degree one function $f=f\left(λ_{1},\cdots,\,λ_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in $\mathbb{R}^{n+1}$, we show that under some suitable conditions on $f$ and the covariant derivati…

We construct a series of finitely presented semigroups. The centers of these semigroups encode uniquely up to rigid ambient isotopy in 3-space all non-oriented spatial graphs. This encoding is obtained by using three-page embeddings of graphs into the product of the line with the cone on three points. By exploiting thr…

A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three cla…