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We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G = \SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of…

Recently it was demonstrated that by adding to the Einstein-Hilbert action a series in powers of the curvature invariants with specially chosen coefficients one can obtain a theory of gravity which has spherically symmetric solutions…

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann…

In this paper we provide new necessary and sufficient conditions for the existence of K\"ahler-Einstein metrics on small deformations of a Fano K\"ahler-Einstein manifold. We also show that the Weil-Petersson metric can be approximated by…

We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on $\mathbb{R}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to…

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for…

We study the Wu metric for the non-convex domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $ 0 < m < 1/2$.…

We first provide a stochastic formula for the Carath\'eodory distance in terms of general Markovian couplings and prove a comparison result between the Carath\'eodory distance and the complete K\"ahler metric with a negative lower curvature…

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $ H^2(\Omega ,\varphi)$ with a plurisubharmonic weight…

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat K\"ahler metric on $D$. Then $h$ is…

Some instanton corrections to the universal hypermultiplet moduli space metric of the type-IIA string theory compactified on a Calabi-Yau threefold arise due to multiple wrapping of BPS membranes and fivebranes around certain cycles of…

K\"ahler-Einstein metrics for polarized families of Calabi-Yau manifolds define a natural hermitian metric on the relative canonical bundle. The fact that the curvature form is equal to the pull-back of the Weil-Petersson form up to a…

We study the completeness of a metric which is related to the Bergman metric of a bounded domain. We provide a criterion for its completeness in the spirit of the Kobayashi criterion for the completeness of the Bergman metric. In particular…

We give a description of all $G$-invariant Ricci-flat K\"ahler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As…

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…

The general class of Robinson-Trautman metrics that describe gravitational radiation in the exterior of bounded sources in four space-time dimensions is shown to admit zero curvature formulation in terms of appropriately chosen…

In this paper, we give the flag curvature formula of general $(\alpha,\beta)$-metrics of Berwald type. We study conformally related $(\alpha,\beta)$-metrics, especially general $(\alpha,\beta)$-metrics that are conformally related to…

We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are…

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved…

In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of $n$-dimensional Riemannian…