Limiting Behavior of a Class of Hermitian-Yang-Mills Metrics

Motivated by mirror symmetry, we begin to study the limiting behavior of Hermitian-Yang-Mills metrics on a class of rank two slope-stable vector bundles over the product of two elliptic curves with a family of product metrics, which are flat and have areas $\epsilon$ and $\epsilon^{-1}$ on two factors respectively. The method is to construct a family of Hermitian metrics and then compare them with the normalized Hermitian-Yang-Mills metrics. We find that the metrics are close in $C^k$ to arbitrary order in $\epsilon$.Comment: Lemma 14 is added, which fixes the C^0 estimat

Relations between the Kahler cone and the balanced cone of a Kahler manifold

In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.Comment: Some corrects have been mad

Twistor geometry of Riemannian 4-manifolds by moving frames

In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary connection on the vertical tangent bundle over the twistor space $Z$, we can recover J. Fine and D. Panov's result on the condition of the first Chern form being symplectic and P. Gauduchon's result on the condition of the first Chern form being a (1,1)-form respectively, by using the method of moving frames

Complex Balanced Spaces

In this paper, the concept of balanced manifolds is generalized to reduced complex spaces: the class B and balanced spaces. Compared with the case of Kahlerian, the class B is similar to the Fujiki class C and the balanced space is similar to the Kahler space. Some properties about these complex spaces are obtained, and the relations between the balanced spaces and the class B are studied.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1610.0715

Form-type Calabi-Yau equations on K\"ahler manifolds of nonnegative orthogonal bisectional curvature

In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on K\"ahler manifolds of nonnegative orthogonal bisectional curvature.Comment: Added a remark in the introductio

Semilinear equations, the γk\gamma_k function, and generalized Gauduchon metrics

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $\gamma_k$ functions on the space of its hermitian metrics.Comment: Some examples and references are added. This version was finished on February 1

On complete constant scalar curvature K\"ahler metrics with Poincar\'e-Mok-Yau asymptotic property

Let $X$ be a compact K\"ahler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature K\"ahler metrics on non-compact K\"ahler manifold $X-S$ with Poincar\'e--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics

Balanced metrics on non-Kahler Calabi-Yau threefolds

We construct balanced metrics on the family of non-K\"ahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on K\"ahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $k\geq 2$ copies of $S^3\times S^3$.Comment: Title changed, introduction rewritten and details of the proof of Lemma 4.2 adde

Scalar curvatures in almost Hermitian geometry and some applications

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we get some inequalities of various total scalar curvatures and some characterization results of the K\"{a}hler metric, balanced metric, locally conformally K\"{a}hler metric and the k-Gauduchon metric. As corollaries, we show some results related to a problem given by Lejmi-Upmeier \cite{LeU} and a conjecture given by Angella-Otal-Ugarte-Villacampa \cite{AOUV}

Lower Bound for the Simplicial Volume of Closed Manifolds Covered by H2×H2×H2\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}

We estimate the upper bound for the $\ell^{\infty}$-norm of the volume form on $\mathbb{H}^2\times\mathbb{H}^2\times\mathbb{H}^2$ seen as a class in $H_{c}^{6}(\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R};\mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $\big(\mathbb{H}^2\big)^n$