arXiv research

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated mini…

Paper connects minimal and maximal surfaces' boundary value problems.

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surfa…

We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…

We prove an analog of the Schoen-Yau univalentness theorem for saddle maps between discs.

New bounds link Schwarzian derivative to hyperbolic geometry.

Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.

The valence of a function $f$ at a point $w$ is the number of distinct, finite solutions to $f(z) = w$. Let $f$ be a complex-valued harmonic function in an open set $R \subseteq \mathbb{C}$. Let $S$ denote the critical set of $f$ and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the com…

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least…

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane doma…

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $φ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a ge…

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouvil…

Study on harmonic maps on weighted Riemannian foliations.

Study cohomology classes related to harmonic maps on submersions.

We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemann…

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

Dirac-harmonic maps are uncoupled under certain conditions.

Extends p-harmonic map theory for new properties.

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

Paper proves existence of smooth nontrivial Dirac-harmonic maps.

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

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

We obtain various estimates of the life-time of two-dimensional minimal tubes in R^3 by potential theory methods.

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

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

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 …

$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between E…

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be the set of points $ξ$ such that $\lim_{z\to ξ}\|F(z)\|=\infty$. Then we prove that…

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

Constructs harmonic maps between special geometric shapes.

Study cohomology classes related to $n$-harmonic morphisms and $F$-harmonic maps.

Harmonic maps studied in sub-Riemannian geometry for Lie groups.

The paper studies geometric properties of $Φ_{(3)}$-harmonic maps and proves Liouville type results.

The paper examines stability of harmonic maps on specific manifolds.

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existe…

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

Develops new methods for Epstein surfaces and W-volume.

The paper examines stability of subelliptic harmonic maps with potential.

Survey on Higgs bundles via harmonic maps and their role in non-abelian Hodge correspondence.

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

Extends Fatou theorem to bounded harmonic maps.

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty$. Infinity harmoncity appears in many familiar contexts. For example,…

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

The paper proves a Liouville theorem for a specific type of harmonic maps on foliated manifolds.

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

Study on stability of harmonic maps with sub-Riemannian geometry.