arXiv research

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…

Geometric limits of cyclic subgroups in specific groups studied.

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove…

Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.

Study describes limits of non-collapsing K3 surfaces using algebraic data.

Constructs knots from 3-manifolds with specified geometric limits.

Study continuity of limit sets in symmetric spaces.

The study finds many tight contact structures on hyperbolic 3-spheres.

Deep learning models complex multivariate extremes using geometric shapes.

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

We describe the topology of the space of all geometric limits of closed abelian subgroups of PSL2C. Main tools and ideas come from the previous paper [BC12].

In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain ba…

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with geometrically finite limit. This allows us to show that the renormalized volume attains its…

Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…

The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.

The large-N limit of Segal-Bargmann transform on spheres is studied.

In this paper we research the differential geometric and algebro-geometric proper- ties of the noncollasping limit in the conical continuity equation.

Paper studies convergence of nonnegative scalar curvature metrics to a specific limit space.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $π_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we construct bi-Lipschitz model manifolds for such hyperbolic 3-manifolds, which have a stru…

Study shows central limit theorem for counting measures in non-smooth spaces.

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions …

In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.

Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into PSL(2,C) does not contain parabolics, then it is also the sequence's geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences…

Geometric tempering fails for Langevin dynamics, proving convergence limits.

In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in \cite{LaTi14}.

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

New method for geometric flows with surgery without smooth estimates.

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Eve…

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geome…

Study on the geometric Dyson Brownian motion of non-square matrix products.

Study geometric properties of surfaces with specific formulae.

The paper explores continuous limits of pentagram maps and their relation to KdV equations.

For a torsion free Kleinian group $Γ$ without parabolics, we consider the decomposition of the limit set $L(Γ)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(Γ)$ when $L(Γ)=S^2_\infty$.

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps.…

We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.

We prove a Lorentzian splitting theorem with weakened curvature conditions.

Paper describes integrable structure of Hitchin moduli spaces.

This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the mat…

Study shows Julia sets and gasket limit sets are quasiconformally different.

Study shows local topologies of certain geometric spaces.

Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…

The paper explores geometric properties of interception curves on planes and spheres.