arXiv research

Explicit presentations found for asymptotically rigid mapping class groups.

Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…

Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.

We prove asymptotic faithfulness for the quantum $\mathrm{Sp}(4)$ mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the $A_n$ family. The methods used are generalized from the proof of asymptotic faithfulness for skein $SU(2)_k$ mapping class group represent…

Proves asymptotic mapping class groups of Cantor manifolds are of type F_infinity.

Classifies 2-uniform maps on torus with formulas and asymptotic bounds.

A Thurston map is a branched covering map from $§^2$ to $§^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on $§^2$ containing $\post(f)$. The boundary at infinity of $\G$ with associated visual me…

Study mapping class groups on CAT(0) cube complexes.

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, …

Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.

We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…

Study on finiteness properties of handlebody mapping class groups.

The study explores normal generators for mapping class groups and their properties.

In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spaces of flat connections on surfa…

Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.

We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.

The study proves properties of intersections of horospheres in harmonic spaces.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the …

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several a…

Solves Dirichlet problem for harmonic maps to give geodesic insights.

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…

We construct sequences of pseudo-Anosov mapping classes whose dilatations behave asymptotically like the inverse of the Euler characteristic of the surface they are defined on. These sequences are used to show that if the genus, g, and punctures, n, of a surface are related by a rational ray g=rn then the minimal dilat…

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal fo…

Geometric models help classify infinite-type surface mapping class groups.

Paper shows how scattering maps of Schrödinger equations relate to metrics.

Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.

We study the family of holomorphic maps from the polydisk to the disk which restrict to the identity on the diagonal. In particular, we analyze the asymptotics of the orbit of such a map under the conjugation action of a unipotent subgroup of $\text{PSL}_2(\mathbb{R})$. We discuss an application our results to the stud…

In this paper, we consider the heat flow for p-pseudoharmonic maps from a closed Sasakian manifold M into a compact Riemannian manifold N. We prove global existence and asymptotic convergence of the solution for the p-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold N is nonpos…

New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.

Study of harmonic maps with extreme Kerr-like singularities.

Estimates point counts in Teichmüller space for mapping class groups.

New estimates for Hitchin's equations at high energy.

Establishes scattering theory for de Sitter vacuum solutions in even dimensions.

The curve graph and related graphs are hyperbolic and have quasi-tree fibers.

Study counts orbits of mapping class group in shearing coordinates.

The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston cell complex whose vertices are asymptotically trivial pants decompositions. We …

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $ψ$. We show that there exists a sequence $(R_n, ψ_n)$ of fibers and monodromies contained in the fibered cone of $(S,ψ)$ such that the asymptotic translation length of $ψ_n$ on the curve complex $\mathca…

ERM performs well in feature learning with minimal feature maps.

The paper generalizes the moment map interpretation of scalar curvature in Kähler geometry.

The paper compares Bayesian uncertainty to MAP estimator in random features regression.

Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.

Given a Riemann surface $X = (Σ, J)$ we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,tω_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tildeΣ,\tilde{J})…

Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.

New groups from strand diagrams show polycyclic subgroups are virtually abelian and undistorted.

We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, ha…