Related papers: Weil-Petersson metric on the universal Teichmuller…

Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichm\"uller spaces for $p \geq 1$ provides the coordinates for the space of $p$-Weil-Petersson embeddings $\gamma$ of the real line $\mathbb R$ into…

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a…

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…

Curvature and torsion are the two tensors characterizing a general Riemannian spacetime. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the…

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichm\"uller space $T_S^{comb}$ equipped with Kontsevich symplectic form $\omega_K$, and then the usual Weil-Petersson geometry of…

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an…

In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact K\"{a}hler manifold. The starting point of our study is…

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M,…

Let $\pi: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact K\"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique K\"ahler-Einstein…

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the…

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $\pi_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichm\"uller…

The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the…

In this paper we continue our study on the canonical metrics on the Teichm\"uller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carath\'eodory metric to the K\"ahler-Einstein metric,…

Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|,…

In this paper we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalizes the twistor space of quaternion-Hermitian manifolds and weak-Spin(9) structures. We also construct almost complex…

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of…

In Riemann geometry, the relations among two transversal submanifolds and global manifold are discussed. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols…

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichm\"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration…

A Teichm\"uller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichm\"uller curves of positive genus. Our methods are based on…

Geometry and topology are fundamental to modern condensed matter physics, but their precise connection in quantum systems remains incompletely understood. Here, we develop an analytical scheme for calculating the curvature of the quantum…