arXiv research

Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.

Pluriharmonic maps form an important class of harmonic maps which includes holomorphic maps. We study their morphisms, in particular the inter-relationships between $(1,1)$-geodesic, pluriharmonic and $\pm$holomorphic maps. Then we characterise pluriharmonic morphisms between Hermitian manifolds. We make a special stud…

Study pseudoholomorphic maps using canonical connection.

The paper broadens a mathematical correspondence to include more balanced metrics.

In this paper, we show that one can interrelate pluriharmonic maps with para-pluriharmonic maps by means of the loop group method. As an appendix, we give examples for the interrelation between pluriharmonic maps and para-pluriharmonic maps. Moreover, we investigate the relation among CMC-surfaces by use of such maps.

As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen from the definition of an anti-Riemannian map. Then we give a decomposition theo…

This paper constructs and proves the uniqueness of pluriharmonic maps to Euclidean buildings.

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to…

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms …

In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …

In this paper, we investigate critical maps of the horizontal energy functional $E_{H,\widetilde{H}}(f)$ for maps between two pseudo-Hermitian manifolds $(M^{2m+1},H(M),J,θ)$ and $(N^{2n+1},\widetilde{H}(N), \widetilde{J},\widetildeθ)$. These critical maps are referred to as $(H,\widetilde{H})$-harmonic maps. We derive…

Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.

Subharmonicity of Dirichlet energy proven for Kähler manifolds.

In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples the source has a non-integrable complex structure, like for instance a nearly Kaeh…

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

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from CR manifolds into Riemannian manifolds or Kahler manifolds. Some basicity, pluriharmonicity and Siu-Sampson type results are established for both harmonic maps and pseudoharmonic maps.

Paper shows CR $Q$-curvature orthogonal to CR pluriharmonic functions.

The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.

Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a re…

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

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

Proves conjecture on deformation invariance of big fundamental groups.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

Paper proves constant functions for pluriharmonic on certain solitons.

The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.

The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.

Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kern…

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establis…

We consider the Johnson-Koranyi-Hua system on symmetric Siegel domains of type two. We prove that all functions which are annihilated by the system and satisfy an H^2 integrability condition are pluriharmonic. So the situation is completely different on type two domains than on tube type domains: it was proved by Johns…

We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian …

In this paper, we study the existence of various harmonic maps from Hermitian manifolds to Kaehler, Hermitian and Riemannian manifolds respectively. By using refined Bochner formulas on Hermitian (possibly non-Kaehler) manifolds, we derive new rigidity results on Hermitian harmonic maps from compact Hermitian manifolds…

In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain …

Study generalizes map properties between Hermitian manifolds preserving specific forms.

This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type …

In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a comapct Hermitian manif…

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

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of…

Study minimal Kähler submanifolds in product of space forms.

We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-$Q$ curvature, these operators d…

On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditio…

Study pseudo-harmonic maps on Weyl manifolds.

For a Kähler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $Δ_{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a finite energy $f$ harmonic …

Maps complex varieties into buildings with harmonic properties.

A Hermitian TQFT from non-semisimple quantum sl(2) modules.

We introduce holomorphic Riemannian maps between almost Hermitian manifolds as a generalization of holomorphic submanifolds and holomorphic submersions, give examples and obtain a geometric characterization of harmonic holomorphic Riemannian maps from almost Hermitian manifolds to Kaehler manifolds.

Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We appl…