arXiv research

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow a…

We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove u…

We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3-sphere, $H$-surfaces in Euclidean 3-space…

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_y:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a res…

Smoothly bounded domains have special functions that are plurisubharmonic.

Let $\varphi\in C^0 \cap W^{1,2}(Σ, X)$ where $Σ$ is a compact Riemann surface, $X$ is a compact locally CAT(1) space, and $W^{1,2}(Σ,X)$ is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map $u:Σ\to X$ homotopic to $\varphi$ or there exists a co…

We discuss non-conformal harmonic surfaces in $R^3$ with prescribed ($\pm$)transforms, and we get a representation formula for non-conformal harmonic surfaces in $R^3$.

This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-f…

In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic maps is smooth and has dimension two. We also show that the points that correspon…

New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.

Harmonic maps intersect all minimal surfaces with bounded curvature.

In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-m…

Harmonic maps from surfaces to CAT(k) spheres are branched coverings.

This is an expository article which describes one approach to the construction and classification of harmonic tori "of finite type", namely, via their ring of polynomial Killing fields. To keep the discussion focussed, the first section is devoted entirely to non-conformal harmonic tori in the 2-sphere. The second sect…

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\R^{n+2}$. On the level of their conformal Gauss maps…

Study bounds the Morse index of a special torus to 1.

We will investigate the local geometry of the surfaces in the $7$-dimensional Euclidean space associated to harmonic maps from a Riemann surface $Σ$ into $S^6$. By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions $\varphi:Σ\to S^6$ whose associated immersio…

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biha…

We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost and Parker respectively establ…

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…

In this article, we use the harmonic sequence associated to a weakly conformal harmonic map $f:S\to S^6$ in order to determine explicit examples of linearly full almost complex 2-spheres of $S^6$ with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry a…

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

The paper discusses Gauss maps for Möbius surfaces in spheres and their applications to Willmore surfaces.

Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.

Sharp estimate on harmonic maps at conformal points in balls.

Paper solves long-standing Gaussian curvature conjecture for minimal graphs.

In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manif…

For odd dimensional Poincaré-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for small $m$ but it becomes finite dimensional if $m$ is large enough, and in one-to-o…

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal …

Holomorphic map connects Hitchin components to character varieties.

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may…

Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic probl…

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective- Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named "weakly Z symmetric" and denoted by (WZS)_n. If the Z tensor is singular we give conditions for the exis…

The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge compani…

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…