Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

In this paper, first we introduce a new notion of commuting condition that $\phi {\phi }_{1}A=A{\phi }_1\phi$ between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$. Suprisingly, real hypersurfaces of type $\left(A\right)$, that is, a tube over a totally geodesic $G_2\left({ℂ}^{m+1}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying the commuting condition. Finally we get a characterization of Type $\left(A\right)$ in terms of such commuting...

We characterize real hypersurfaces with constant holomorphic sectional curvature of a non flat complex space form as the ones which have constant totally real sectional curvature.

We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.

In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in {\mathbb{Z}}^{+}$, i.e., $${\Omega }_n\left(\gamma \right)=\{z=(z_1,\cdots ,z_n)\in {ℂ}^n:|z_1{|}^{1/\gamma }<|z_2|<\cdots <|z_n|<1\}$$ equipped with a natural Kähler form ${\omega }_{g\left(\mu \right)}:=\frac{1}{2}(i/\pi )\partial \overline{\partial }{\Phi }_n$ with $${\Phi }_n\left(z\right)=-{\mu }_{1}ln\left(\right|z_2{|}^{2\gamma }-\left|z_1{|}^2\right)-\sum _{i=2}^{n-1}{\mu }_{i}ln\left(\right|z_{i+1}{|}^2-\left|z_i{|}^2\right)-{\mu }_{n}ln(1-|z_n{|}^2),$$ where $\mu =({\mu }_1,\cdots ,{\mu }_n)$, ${\mu }_i>0$, depending on $n$ parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for $({\Omega }_n\left(\gamma \right),g\left(\mu \right))$ and we prove that $g\left(\mu \right)$ is balanced if and only if ${\mu }_1>1$ and $\gamma {\mu }_1$ is an integer, ${\mu }_i$ are integers such that ${\mu }_i\ge 2$ for all $i=2,...,n-1$, and ${\mu }_n>1$. Second, we prove that $g\left(\mu \right)$ is Kähler-Einstein if and only if ${\mu }_1={\mu }_2=\cdots ={\mu }_n=2\lambda$, where $\lambda$ is a nonzero...

Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.