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We study the Seiberg–Witten (SW) equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the SW equations for any metric which is "asymptotic" to a Poincaré type metric at infinity. Then we compute a lower bound for the L²-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

The warped product structures of Finsler metrics are studied in this paper. We give the formulae of the flag curvature and Ricci curvature of these metrics, and obtain the characterization of such metrics to be Einstein. Some Einstein Finsler metrics of this type are constructed.

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds $M_n$ obtained by blowing up the diagonal of the product of two copies of a $C{P}^n$.

In this paper, we study Finsler metrics expressed in terms of a Riemannian metric, a 1-form, and its norm and find equations with sufficient conditions for such Finsler metrics to become Ricci-flat. Using certain transformations, we show that these equations have solutions and lead to the construction of a large and special class of Einstein metrics.

It is shown that there is a unique Yamabe representative for a generic set of conformal classes in the space of metrics on any manifold. At such classes, the scalar curvature functional is shown to be differentiable on the space of Yamabe metrics. In addition, some sufficient conditions are given which imply that a Yamabe metric of locally maximal scalar curvature is necessarily Einstein.

A quadruple of Lie groups $(G,L,K,H)$, where $G$ is a compact semisimple Lie group, $H\subset K\subset L$ are closed subgroups of $G$, and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces $G/H$, $G/K$ and $G/L$ are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple $(G,L,K,H)$ in our list, we can produce new non-naturally reductive Einstein metrics on the coset space $G/H$, by scaling the Killing form metrics along the complement of $𝔥$ in $𝔨$ and along the complement of $𝔨$ in $𝔩$. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

Exact solutions of Einstein field equations invariant for a non-Abelian bidimensional Lie algebra of Killing fields are described. Physical properties of these gravitational fields are studied, their wave character is checked by making use of covariant criteria and the observable effects of such waves are outlined. The possibility of detection of these waves with modern detectors, spherical resonant antennas in particular, is sketched.

We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided.

Classical metrics (Eguchi–Hanson, Taub–NUT, Fubini–Study) satisfy a certain system of ODE that we introduced. By studying this system we found new Ricci-flat incomplete Riemannian metric in explicit formula and described its behavior at infinity and in the singular point.

A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on the Stiefel manifolds SO(2k+ l)/SO(l).

We consider a general hyper-Kähler metric in dimension 4 with a S¹-action compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of the S¹-monopole coming from the twistor space of the metric.