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…

Harmonic maps between pinched Hadamard surfaces are quasi-conformal.

Method computes harmonic and conformal maps from point clouds.

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

Paper studies heat flow for maps on manifolds, avoiding singularities.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particul…

Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.

New heat flow for harmonic maps avoids singularities but not bubbles.

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. …

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…

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)…

Study on stability of α-harmonic maps and their applications.

Sharp estimate on harmonic maps at conformal points in balls.

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…

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

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…

f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map betwe…

Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

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

The paper explores $\alpha-$harmonic maps and their stability, proving key properties and conditions.

Smoothly bounded domains have special functions that are plurisubharmonic.

This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, tho…

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…

Paper studies smoothness of bi-conformal heat flow on 4-manifolds.

We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image formation for pairs of points in such neighbourhoods.

In this note, we show that some F-harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in [15] and [17] for harmonic and p-harmonic maps.

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetz…

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces.

Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.

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

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

The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.

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

We characterise the maps into the space of $2$-spheres in $S^n$ that are the conformal Gauss maps of conformal immersions of a surface. In particular, we give an invariant formulation and efficient proof of a characterisation, due to Dorfmeister--Wang \cites{DorWan13,DorWan}, of the harmonic maps that are conformal Gau…

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 …

Harmonic maps intersect all minimal surfaces with bounded curvature.

Study on new metrics on para-Kähler-Norden manifolds with conformal deformation.

Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.

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…

We construct explicit examples of Dirac-harmonic maps $(φ, ψ)$ between Riemannian manifolds $(M,g)$ and $(N,g')$ which are non-trivial in the sense that $φ$ is not harmonic. When $\dim M=2$, we also produce examples where $φ$ is harmonic, but not conformal, and $ψ$ is non-trivial.

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is even dimensional. We then build a conformal invariant functional for the maps betw…

We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of integrability of Jacobi fields along harmonic maps from the 2-sphere to the comp…

Unified approach to Laplace and Steklov eigenvalues via $n$-harmonic maps.

In this paper we generalize harmonic maps and morphisms to the \emph{degenerate semi-Riemannian category}, in the case when the manifolds $M$ and $N$ are \emph{stationary} and the map $φ:M\to N$ is \emph{radical-preserving}. We characterize geometrically the notion of \emph{(generalized) horizontal (weak) conformality}…