Related papers: On the balanced condition for the Eguchi-Hanson me…

We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that…

In the framework of the Standard Model the mass of the physical Higgs boson is an arbitrary parameter. In this note we examine whether it is possible to determine the ratio of $m_H /M$, where $M$ denotes any other mass in the theory, such…

Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the…

A twisted Higgs bundle on a K\"ahler manifold $X$ is a pair $(E,\phi)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $\phi\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were…

Let $n\ge 25$ be an integer. In this paper, we construct a smooth metric $g_{0}$ on $\mathbb{S}^n$ with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient…

We give un upper bound Ent(\Omega, g)<\lambda\ of the diastatic entropy Ent(\Omega, g) of a complex bounded domain (\Omega, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric \lambda g. When (\Omega, g)…

We consider the blowup of a point of a compact K\"ahler manifold and a metric of the form $\mu^*h + t b$ on it, where $h$ is a K\"ahler metric on the original manifold and $b$ is Hermitian form that looks like the Fubini--Study metric near…

We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with…

We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are K\"{a}hler. The type II metrics are ALH whose total energy can be negative.

We prove that total instability is a generic phenomenon in the real analytic class of electromagnetic Lagrangian systems under a weak magnetism hypothesis. The main object in the proof is an adaptation of the McGehee blowup for these…

Let $M$ be a compact K\"ahler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete K\"ahler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$.…

In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $\rho$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More…

In this paper, we investigate the blow-up phenomenon of the $H^2$ norm of solutions to the inhomogeneous biharmonic Schrodinger equation in two distinct scenarios. First, we consider the case of negative energy, analyzing separately the…

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter \alpha, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of \alpha, we…

Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of…

For a smooth manifold $M$ we define the Teichm\"uller space $\cT(M)$ of all Riemannian metrics on $M$ and the Teichm\"uller space $\cT^\epsilon(M)$ of $\epsilon$-pinched negatively curved metrics on $M$, where $0\leq\epsilon\leq\infty$. We…

An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is holomorphically…

We study homogeneous Einstein metrics on indecomposable non-K\"ahlerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the…

We study the characteristic structure of the Einstein-Hilbert (EH) action when modifications of the form of $R^2,~ R_{\mu\nu}^2$, $R_{\mu\nu\rho\sigma}^2$ and $C_{\mu\nu\rho\sigma}^2$ are included. We show that when these quadratic terms…

This paper was withdrawn by the author due to an error in the proof of the main result; essentially the parameter R used in the proof may depend on the manifold (M, g), not just on dimension and pinching constant.