arXiv research

Study on conformal harmonic coordinates on manifolds, proving existence and properties.

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. …

Study on 3-manifolds finds regular conformal metrics for rough metrics.

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and wh…

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$…

We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq …

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconform…

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determi…

We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2π$, determined by…

New examples of weakly Einstein conformal products are constructed.

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

In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the Hölder regularity of the compactified metric. To thi…

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.

Proves harmonic coordinates for weak immersions in even dimensions.

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 $R^3$.

Method computes harmonic and conformal maps from point clouds.

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

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vecto…

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

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Analytic plane curves determine unique conformal coordinates.

Harmonic basis vector fields on surfaces

This study recovers electromagnetic parameters on boundaries from impedance and admittance data.

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

Paper studies Laplace operator estimates in harmonic map heat flows.

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

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…

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we …

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

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)…

Constructs examples of centrally harmonic spaces and shows they are not generically harmonic.

Study on Lie groups' conformal foliations and harmonic morphisms.

The paper studies steady solitons with curvature decay and proves their smoothness.

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

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that eve…

Sharp estimate on harmonic maps at conformal points in balls.

We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.

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

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…

We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold M to a two-dimensional conformal manifold N can be, locally, `extended' to a unique harmonic morphism from the heaven space of M to N.

Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.

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…

Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators …

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…

Researchers create non-degenerate harmonic functions on n-dimensional space.

Study of Demoulin surfaces using Gauss maps and conformal coordinates.

We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature $(1,1)$. We will investigate the properties of the solutions of these equations and try to relate the conformal invariant dimension of the space of harmonic and twistor spin…