arXiv research

Smooth maps show Gromoll filtration for spheres.

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new…

Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $φ:\s^3\to \s^3$. We show $φ$ to be unstable and estimate its biharmonic index and nullity. Resolving the s…

Study on generalized ξ-parallel maps in Riemannian geometry.

The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.

The study finds abundant normal generators for mapping class groups.

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 …

Study cohomology rings of 3D manifolds with round fold maps into the plane.

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give a…

The Hilbert map's image is discussed, showing when it's surjective.

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…

Researchers describe a new Thom form for mapping cones.

New descriptions of a subgroup in mapping class groups.

Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the …

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.

Extends harmonic maps compactification to punctured Riemann surfaces.

Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.

Study shows mapping class group dimension for surfaces with punctures.

Method computes harmonic and conformal maps from point clouds.

A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.

We give a detailed discussion about existence and uniqueness of Lu's momentum map. More precisely, we introduce the infinitesimal momentum map, and we study its properties. This allows us to describe the theory of reconstruction of the momentum map from the infinitesimal one. We provide the conditions for the uniquenes…

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

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

The identity map of certain Einstein manifolds is stable in both energy and bienergy.

The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$ of a closed manifold is isotopic to a map realizing the Nielsen number of $f$, which is a lower bound for the number of fixed points among all maps homotopic to $f$. The main theorem of this paper proves this conjecture for all orientation pre…

New mirror maps improve PMD performance in reinforcement learning.

Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there …

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto …

Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.

Euler and Delisle developed a map method for the Russian Empire, which is now outperformed by the Lambert conformal conical projection.

Embeddings of mapping tori for end-periodic graph maps are proven.

New algorithm constrains SOMs to create supervised low-dimensional mappings.

New framework formalizes estimating valid transport maps, revealing their statistical limits.

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic m…

Defines cobordism maps connecting Khovanov and instanton homologies.

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

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

A polyhedral map is called $\{p, q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In 1983, it was shown that there exist infinitely many geometrically realizable $\{p, q\}$-equivelar polyhedral maps if $q > p = 4$, $p > q = 4$ or $q - 3 > p = 3$. It was shown in 2001 that there exist infi…

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…

We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bu…

Establishes jet transversality for regular maps from flexible manifolds.

Paper disproves Wright's periodic map conjecture.

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extensi…

Paper extends cobordism maps in Khovanov and instanton homologies.

Derives the derivative of the Riemann-Hilbert map for surface connections.

3D manifolds can map to a plane with specific curve patterns.