arXiv research

Study calculates Kulkarni limit sets for quaternionic projective groups.

If $Γ$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(Γ)$ of the natural action of $Γ$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(Γ)$ is discontinuous, and $Eq(Γ)$ agrees with the discontinuity set in the sense of Kulkarni whenever the limit s…

We give a topological description of the quotient space $Ω(G)/G$ in the case $G \subset PSL(3, \mathbb{C})$ is a discrete subgroup acting on $\mathbb{P}^2_\mathbb{C}$ and the maximum number of complex projective lines in general position contained in Kulkarni's limit set, $Λ(G)$, is 4. We also give a topological descri…

To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all $θ\in[(n-1)π/2,nπ/2[$ and for all $r>\opTan(θ/n)$ a unique immersed hypersurface $Σ_{r,θ}=(M,i_{r,θ})$ in $\Bbb{H}^{n+1}$ of constant $θ$-special Lagrangian curvature equal to $r$. We show that these hypers…

The paper proves geometric bordisms for specific hyperbolic surfaces.

Some curvature properties of Kahler manifolds of indefinite metrics are studied. Analogues of a Kulkarni's theorem are proved for such manifolds.

We show that a finite dihedral group does not act pseudofreely and locally linearly on a 2k-dimensional sphere, if k > 1. This answers a question of R. S. Kulkarni from 1982.

Conditions, related to Kulkarni's equivalence problem are considered for indefinite Riemannian and Kaehlerian manifolds. Corresponding theorems are obtained for the values of the Ricci tensor on isotropic vectors as well as for the values of the curvature tensor on degenerate holomorphic 2-planes.

Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is co…

Defines a new tensor related to special geometric spaces.

As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the r…

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenböck formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also gi…

Einstein's equation, in its standard form, breaks down at the Big Bang singularity. A new version, equivalent to Einstein's whenever the latter is defined, but applicable in wider situations, is proposed. The new equation remains smooth at the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model. It is…

Paper introduces new center of mass for flat manifolds.

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, oc…

Study curvature properties in special manifolds using specific tensors.

Revisits the Gauss-Bonnet formula using double forms.

Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is greater than 30, the same phenomenon happens for periodic maps on the surfaces wit…

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the expl…

We consider surface branch data with base surface the sphere, odd degree d, three branching points, and two partitions of d of the form (2,...,2,1) and (2,...,2,2h+1). If the third partition has length L, this datum satisfies the Riemann-Hurwitz necessary condition for realizability if h-L is odd and at least -1. For s…

The paper proves a free product decomposition for congruence subgroups with constraints.

Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the p…

Study of hypersurfaces in curved spaces with specific curvature properties.

The curvature of Gauss maps for flat submanifolds is studied in space forms.

Study Wintgen ideal submanifolds in curved spaces with specific curvature conditions.

New algorithm for Coxeter connections with maximally ramified singularities.

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then…

Study shows Julia sets and gasket limit sets are quasiconformally different.

Find limiting sets for digital cones and suspensions.

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…

Study shows non-symmetric convex sets have full boundary limits.

Rare Teichmüller disks converge to small limit sets.

The paper characterizes subgroup stability via limit sets on the Morse boundary.

Study continuity of limit sets in symmetric spaces.

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove…

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) th…

We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being …

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

The paper connects geodesic flows and limit sets on visibility manifolds.

Study contractibility of boundaries in convex sets and limit sets of subgroups.

Study shows spectral gaps limit points on surfaces.

This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.

We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

For a torsion free Kleinian group $Γ$ without parabolics, we consider the decomposition of the limit set $L(Γ)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(Γ)$ when $L(Γ)=S^2_\infty$.

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

Two groups with specific limit sets in hyperbolic spaces are identified.

We relate the L^2 cohomology of a complete hyperbolic manifold to the invariant currents on its limit set.