Jianming Wan
In this paper, we shall prove that a harmonic map from $\mathbb{C}^{n}$ ($n\geq2$) to any Kahler manifold must be holomorphic under an assumption of energy density. It can be considered as a complex analogue of the Liouville type theorem for harmonic maps obtained by Sealey.
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We prove a refined Kato inequality for closed and coclosed differential $(p,q)$ forms on a Kahler manifold.
Almost-Kähler four manifolds with harmonic self-dual Weyl curvature
本文研究具有调和自对偶Weyl曲率的紧致近Kähler四维流形，证明了在常数量曲率且$c_{1}\cdotω\geq 0$条件下该流形为Kähler流形，并建立了一个积分曲率不等式。
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein man…
In this article, we study the $L^{2}$-harmonic forms on the complete $2n$-dimensional almost Käher manifold $X$. We observe that the $L^{2}$-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can extend vanishing theorems of $d$(bounded) (resp. $d$(sublinear)) Kähler manifold proved by Gromov (resp. Cao-Xavier, Jost-Zuo) to almost Kählerian case, that is, the spaces of all harmonic $(p,q)$-forms on $X$ vanishing unless $p+q=n$. We also give a lower bound on the spectra of the Laplace operator to sharpen the Lefschetz vanishing theorem on $d$(bounded) case.
This report attempts a clean presentation of the theory of harmonic maps from complex and Kähler manifolds to Riemannian manifolds. After reviewing the theory of harmonic maps between Riemannian manifolds initiated by Eells--Sampson and the Bochner technique, we specialize to Kähler domains and introduce pluriharmonic maps. We prove a refined Bochner formula due to Siu and Sampson and its main consequences, such as the strong rigidity results of Siu. We also recount the applications to symmetric spaces of noncompact type and their relation to Mostow rigidity. Finally, we explain the key role o…
Let $(X,J,g,ω)$ be a compact $2n$-dimensional almost-Kähler manifold. We prove primitive decompositions of $\partial$-, $\overline{\partial}$-harmonic forms on $X$ in bidegree $(1,1)$ and $(n-1,n-1)$ (such bidegrees appear to be optimal). We provide examples showing that in bidegree $(1,1)$ the $\partial$- and $\overline{\partial}$-decompositions differ.
Clairaut semi-slant/hemi-slant Riemannian maps to Kähler manifolds
本文研究了Clairaut半斜和半斜Riemannian映射到Kähler流形的性质与特性。
We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kähler moment map $ϕ$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $ϕ$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the comple…
In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler manifolds.
Refined Kato type inequalities and new vanishing theorems on complete Kähler and quaternionic Kähler manifolds
本文利用Böchner技巧与改进的Kato型不等式，将消没定理推广至完备Kähler与四元数Kähler流形上的$L^p$多重调和函数及调和1-形式，并证明Sobolev型不等式下的有限$L^p$能量多重调和函数消没。
Exploring certain geometric and harmonic properties of the Berger-type metric conformal deformation on the Para-Kähler-Norden manifold
该研究在Para-Kähler-Norden流形上构造了由Berger型度量共形形变得到的新度量类，分析了其Levi-Civita联络与各类曲率，并研究了相关的一类调和映射。
On $\overline{\partial }_{b}$-harmonic maps from pseudo-Hermitian manifolds to Kähler manifolds
本文研究从伪埃尔米特流形到Kähler流形的映射，引入偏能量泛函，定义并探讨了$\overline{\partial }_{b}$-调和映射与$\partial _{b}$-调和映射，推广了叶状Lichnerowicz型结果及Siu的全纯性结论。
We prove a sharp integral gradient estimate for harmonic functions on noncompact Kähler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.
Conformal pointwise slant Riemannian maps from or to Kähler manifolds
本文研究从或到Kähler流形的共形逐点斜Riemann映射，提供非平凡例子，推导重要结果，讨论分布的可积性与全测地性，并研究其同伦性、调和性条件及不等式。
Let $(X,J,ω)$ be a compact $2n$-dimensional almost Kähler manifold. We prove primitive decompositions for Bott-Chern and Aeppli harmonic forms in special bidegrees and show that such bidegrees are optimal. We also show how the spaces of primitive Bott-Chern, Aeppli, Dolbeault and $\partial$-harmonic forms on $(X,J,ω)$ are related.
In this article, we study harmonic symmetries on the compact locally conformally Kähler manifold $M$ of $dim_{\mathbb{C}}=n$. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator $\square$. We observe that the spaces $\ker(\square)\capΩ^{l}=\{0\}$ for any $|l-n|\geq2$ and $\kerΔ_{\bar{\partial}}\cap P^{k,n-1-k}\cap\ker(i_{θ^{\sharp}})\cong\ker(\square^{k,n-1-k})$, $\kerΔ_{\bar{\partial}}\cap P^{k,n-k}\cong\ker(\square^{k,n-k})$. Furthermore, suppose that $M$ is a Vaisman manifold, we prove that (i) $α$ is…
Clairaut Generic Riemannian Maps from Nearly Kahler Manifolds
本文研究了从近卡氏流形到黎曼流形的克莱尔aut 通用黎曼映射，并获得该映射为总流形上全测地叶的条件，还给出了非平凡的例子。
In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.
Harmonic flow of quaternion-Kähler structures
本文建立 8 维流形上 $Sp(2)Sp(1)$-结构的梯度 Dirichlet 流理论，证明近单调性、小能量长时间存在性，并构造调和 QK 孤立子及显式收敛解。
We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.
We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that…
In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.
We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,ω)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $ψ\in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $ω^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provi…
On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Bochner type formula reflecting the Kahler structure.
The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of $d$-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler $4$-manifolds. In particular, these provide topological bounds on the dimension of the space…
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 into any Kaehler manifolds and harmonic maps into Kaehler manifolds with strongly semi-negative curvature respectively. These monotonicity inequalities enable us to derive some holomorphicity and Liouville type results for these pluriharmonic maps and harmonic maps. We also use t…
We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
Clairaut slant Riemannian maps to Kähler manifolds
本文研究从黎曼流形到Kähler流形的Clairaut斜黎曼映射，推导了多类相关充要条件，分析了其调和性，得到两个第二基本形式相关不等式及等号情形。
We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kahler-Ricci flow.
In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact Kähler manifolds.
Using the harmonic superspace formalism, we find the metric of a certain 8-dimensional manifold. This manifold is not compact and represents an 8-dimensional generalization of the Taub-NUT manifold. Our conjecture is that the metric that we derived is equivalent to the known metric possessing a discrete $Z_2$ isometry, which may be obtained from the metric describing the dynamics of four BPS monopoles by Hamiltonian reduction.
A vanishing result for harmonic $(n-1,1)$-forms
本文证明闭凯勒流形在特定正曲率条件下本原调和$(n-1,1)$-型消失，进而推出其必为复射影空间。
We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. For example, we show that no co-compact lattice in SO(1,n), n>2, can be the fundamental group of a compact Kahler-Weyl manifold of certain type.
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 function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for $f$-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such $f$-harmonic maps must be constant if $f$ has an isolated minimum point. In pa…
We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the Hermitian metric on the compact almost complex manifold is strongly Gauduchon. Under the further assumption that the almost complex structure is integrable, we show that the integral condition for a Gauduchon metric is equivalent to be strongly Gauduchon. In particular, a compact complex surface with a Gauduchon metric satisfying the integral condition is automatica…
We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give necessary and sufficient conditions for totally geodesicness and harmonicity of this type submersions. Moreover, we investigate the several curvatures of the total manifold and fibers and give a characterization theorem.
We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.
Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).
Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is defined on the Kähler class, correspond to metrics with harmonic $k$th Chern form. In this paper it is shown that the higher K-energy functionals, when restricted to the Bergman metrics, are expressible as the energies of certain pairs of vectors (tensors products of discriminants).…
In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.
In this paper, we study the existence of harmonic and bi-harmonic maps into Riemannian manifolds admitting a conformal vector field, or a nontrivial Ricci solitons.
In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the Schwarz lemma for transversally harmonic maps of bounded generalized transversal dilatation. In addition, we also obtain a Schwarz type lemma for transversally holomorphic maps between Kähler foliations.
J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.
$\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds
研究四维紧近复流形上埃尔米特度量的$\overline\partial$-调和(1,1)-形式维数，证明全局共形凯勒时维数为$b_-+1$，局部共形凯勒时维数为$b_-$，并给出Kodaira-Thurston流形上的例子表明该维数依赖于度量。
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 concepts of harmonic maps and bi-harmonic maps. We first derive the $f$-biharmonic map equation and then use it to study $f$-bi-harmonicity of some special maps, including conformal maps between manifolds of same dimensions, some product maps between direct product manifold and s…
In this note, we investigate estimates of the Morse index for F-harmonic maps into spheres, our results extend partially those obtained in ([14]) and ([15]) for harmonic and p-harmonic maps.
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 and relies on results of Gursky-Viaclovsky and Lamm.
In this paper, we extend the definition of p-harmonic and p-biharmonic maps between Riemannian manifolds. We present some new properties for the generalized stable p-harmonic maps.
$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $α$-harmonic maps for $α>1$ and then letting $α\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $α$-Dirac-harmonic maps when …
On harmonic and biharmonic maps from gradient Ricci solitons
该研究聚焦梯度Ricci孤立子上的调和与双调和映射，推导了使调和映射为常值、双调和映射退化为调和映射的解析与几何条件，并证明二维cigar孤立子的有限能量双调和映射必为调和映射。
The CR analogue of B.-Y. Chen's conjecture on pseudo biharmonic maps will be shown. Pseudo biharmonic, but not pseudo harmonic, isometric immersions with parallel pseudo mean curvature vector fields, will be characterized. Several examples of pseudo biharmonic maps will be given.
A Framework for Gluing Harmonic Maps
本文提出一个粘合框架，在特定条件下构造从黎曼曲面到黎曼流形 $N$ 的扩张调和映射的粘合映射。
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.
We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.
Construction of harmonic maps between cohomogeneity one manifolds
本文提出了一种构造cohomogeneity one manifolds之间等变调和映射的方法。
J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we give the second variation formula of k-harmonic maps, and show non-existence theorem of proper k-harmonic maps into a Riemannian manifold of non-positive curvature (k >= 2). We also study k-harmonic maps into the product Riemannian manifold, and describe the ordinary differential equations of 3-harmonic curves and 4-harmonic curves into a sphere, and show their non-trivial solutions.
This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly Kähler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an $\mathrm{SU}(2)$ subgroup of nearly Kähler automorphisms is automatically homogeneous.
Bounded harmonic maps
该研究将经典的Fatou定理从单位圆盘上的有界调和函数推广到有界调和映射的情况，拓展了相关定理的适用范围。
We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.
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