Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCongruence classes of points in quaternionic hyperbolic spaces
57
0
0.0
(
0
)
Authors
Wensheng Cao
Ask ChatGPT about the research
No Arabic abstract
An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $overline{{bf H}_bh^n}$, up to congruence in the holomorphic isometry group ${rm PSp}(n,1)$ of ${bf H}_bh^n$. In this paper we concentrate on two cases: $m=3$ in $overline{{bf H}_bh^n}$ and $m=4$ on $partial{bf H}_bh^n$ for $ngeq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartans angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.
rate research
Read More
Let $mathcal{M}(n,m;F bp^n)$ be the configuration space of $m$-tuples of pairwise distinct points in $F bp^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in $F bp^n$ with respect to the diagonal action of ${rm PU}(1,n;F)$ equipped with the quotient topology. It is an important problem in hyperbolic geometry to parameterize $mathcal{M}(n,m;F bp^n)$ and study the geometric and topological structures on the associated parameter space. In this paper, by mainly using the rotation-normalized and block-normalized algorithms, we construct the parameter spaces of both $mathcal{M}(n,m; bhq)$ and $mathcal{M}(n,m;bp(V_+))$, respectively.
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb R^n$, $nge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashis pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb R^n$, this minimal metric coincides with the classical Beltrami-Cayley-Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a domain having a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. Finally, we show that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-plane.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
We prove that both the Laplacian on functions, and the Lichnerowicz Laplacian on symmetric 2-tensors with respect to asymptotically hyperbolic metrics, are sectorial maps in weighted Holder spaces. As an application, the machinery of analytic semigroups then applies to yield well-posedness results for parabolic evolution equations in these spaces.
Let $M$ be a complex manifold. We prove that a compact submanifold $Ssubset M$ with splitting tangent sequence (called a splitting submanifold) is rational homogeneous when $M$ is in a large class of rational homogeneous spaces of Picard number one. Moreover, when $M$ is irreducible Hermitian symmetric, we prove that $S$ must be also Hermitian symmetric. The basic tool we use is the restriction and projection map $pi$ of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with $dimgeq 2$ in a hyperquadric, which has been previously proven using algebraic geometry.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
