arXiv research

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations a…

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.

Estimates for geodesics on hyperbolic tori improve previous bounds.

We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eight possible types, exactly one consists of geodesics that form a bigon about the cusp; we express all such geodesics in terms of Markoff trip…

Study trace systoles on surfaces, finding optimal bounds and implications.

We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the description of a non-empty open subset of C^n on which the group acts properly discon…

This paper has been withdrawn by author due to an error in the proof.

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…

Research extends geodesic length function study to three holed sphere.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

A musical instrument based on moduli spaces lets users hear geometric concepts.

The article explores the mapping class group using unicellular maps and provides filtrations.

Constructs a moment map flow for isotropic maps on surfaces.

Maps are an important medium that enable people to comprehensively understand the configuration of cultural activities and natural elements over different times and places. Although massive maps are available in the digital era, how to effectively and accurately access the required map remains a challenge today. Previo…

The paper constructs biharmonic maps between spheres using polynomial maps.

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy functional $\frac{1}{2}\int_M f|τ(φ)|^2dv_{g}$. This class of maps generalizes both …

Research explores real algebraic realization of round fold maps of codimension -1.

The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.

This paper shows semi-equivelar toroidal maps are vertex-transitive covers.

Paper defines and studies Clairaut warped product Riemannian maps.

The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.

The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…

This paper constructs real algebraic maps that are topologically special generic maps.

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…

The paper constructs gluing maps for harmonic maps between Riemannian manifolds.

Characterizes a general range decreasing group homomorphism.

The paper examines $HM$-tensional and $HS$-tensional maps between Riemannian manifolds.

The paper proves a Liouville theorem for specific harmonic maps with free boundary.

Dirac-harmonic maps are uncoupled under certain conditions.

We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficie…

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.

In this paper, we show that one can interrelate pluriharmonic maps with para-pluriharmonic maps by means of the loop group method. As an appendix, we give examples for the interrelation between pluriharmonic maps and para-pluriharmonic maps. Moreover, we investigate the relation among CMC-surfaces by use of such maps.

The article explores constructing biharmonic and conformal biharmonic maps to spheres.

Harmonic map flow preserves almost-holomorphic maps without singularities.

Method computes harmonic and conformal maps from point clouds.

New theorem proves convergence of various discrete conformal structures to conformal maps.

Constructs harmonic maps between special geometric shapes.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

Study isotropic Riemannian maps and helices along them.

Study on harmonic maps on weighted Riemannian foliations.

Maps and measures on surfaces link best Lipschitz and least gradient functions.

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …

Paper proves existence and gives construction of Symphonic map between ellipsoids.

Study heat flow for half-harmonic maps and harmonic maps with free boundary.

Paper proves symphonic map result similar to Eells-Sampson.

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse…