arXiv research

In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and fine…

The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.

We use twistor theory to identify the harmonic hull of an arbitrary connected open subset U of R^{2m} for m at least 2. It is the natural domain of analytic continuation in C^{2m} for harmonic functions on U.

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

Extends p-harmonic map theory for new properties.

Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.

Extends harmonic map theory to arbitrary surfaces.

Improved optimal regularity for harmonic almost complex structures.

Study harmonic mappings and submanifolds using Bochner technique.

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

The paper characterizes harmonic maps and studies their properties on Riemannian manifolds.

The report presents the theory of harmonic maps from Kähler manifolds.

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…

Notes on harmonic maps between manifolds, existence and regularity covered.

Study metric perturbations to make degenerate harmonic forms non-degenerate.

The paper is devoted to the study of the global geometries of harmonic mappings and infinitesimal harmonic transformations and presents their applications to the theory of Ricci solitons.

Scattering theory for harmonic one-forms on Riemann surfaces.

The paper studies harmonic map flows and proves rectifiability of singular sets.

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …

We present a pair of open smooth $4$-manifolds that are mutually homeomorphic. One of them admits a Riemannian metric that possesses quasi-cylindricity, and positivity of scalar curvature and of dimension of certain $L^2$ harmonic forms. By contrast, for the other manifold, no Riemannian metric can simultaneously satis…

New mathematical tools for studying knots and links.

Develops potential theory for WZW equation in Kähler potentials space.

BacHMMachine harmonizes Baroque chorales using theory-driven principles and Hidden Markov Models.

This paper shows how Hodge's theory of harmonic $p$-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author's definition of flag vectors…

We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold…

Solves initial value problem for harmonic maps on specific manifolds.

In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structu…

This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from the two-sphere to a compact Lie group or symmetric space. Extended solutions are e…

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizon and Chmaj. This sequence shares many features of the Bartnik-McKinnon sequence of solutions to the Einstein-Yang-Mills equations as well as sequences of solutions that have arisen in other physical models. We ap…

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos-Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theor…

Harmonic maps study on surfaces with non-positive curvature.

Complex-valued (p,q)-harmonic morphisms defined and studied.

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.

We study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic …

We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.

Paper proves existence of Dirac-harmonic maps with trivial index.

Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci tensor and the section curvature of $(\bar{M},\bar{g})$ is nonpositive. Moreover, other …

New theorem on flat tori stability using harmonic maps and Ricci flow.

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angl…

Study proves rigidity of harmonic maps from 2-torus to complex projective space.

Study improves regularity estimates for harmonic maps into ellipsoids.

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In thi…

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

Harmonic maps depend analytically on representations.