Related papers: $\mathbb{T}^n$-invariant Kaehler-Einstein manifold…

We prove that any simply connected special Kaehler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we…

In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, i.e. the Torelli space, is a complex affine manifold. As applications we prove that the period map from the…

A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous…

The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts.…

An outstanding open question, which has attracted renewed attention following the pioneering work of Huang--Li--Treuer, is whether, for a given positive integer $m$, there exists a complex manifold whose Bergman metric is locally isometric…

We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of…

In this article, we study quasi-Einstein manifolds with constant scalar curvature. We provide a classification of compact and noncompact (possibly with boundary) $T$-flat quasi-Einstein manifolds with constant scalar curvature, where the…

This is an exposition of results on the existence problem of $\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are…

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds…

Starting from the classification of real Manin triples done in a previous paper we look for those that are isomorphic as 6-dimensional Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several…

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…

We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of…

Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…

Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians $G_{m}(\mathbb{C}^{n+m})$ with $m,n\geq 2$. The…

The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum…

On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes'…

This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous…

We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not…