arXiv research

Study on parabolic points and cylindrical surfaces in Euclidean 3-space.

Defines connections on parabolic vector bundles for Lie algebroids.

In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we give a theorem of Viro's patchworking type to build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. We use this result to build a family …

Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.

The study classifies points on ruled surfaces in 4-space based on geometric properties.

We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the p…

We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…

We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries - in parti…

Real slices of parabolic opers on Riemann surfaces are studied.

We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.

The abstract proves the non-existence of certain real algebraic surfaces.

We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups witho…

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion--free and some types of them have to be locally flat. We prove that each symmetry given at a point with non--zero harmonic curvature is involutive. Finally we giv…

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic sta…

The paper proves stability of pulled back parabolic bundles on curves.

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given …

Computes deformations of parabolic structures on Riemann surfaces.

We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.

Reproves results on spherical metrics using parabolic bundles.

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In pa…

New estimates for nodal and singular sets of parabolic inequalities.

Study gauge theory of real and quaternionic parabolic bundles over real curves.

Study of parabolic Higgs bundles on curves with special fixed points.

The paper studies groups formed by two parabolic maps and their properties.

Consider the moduli space of parabolic Higgs bundles (E,Φ) of rank two on CP^1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E,Φ)\mapsto (E,-Φ). We study the fixed point locus of this involution. In [GM], this modu…

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. This extends earlier work don…

We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization procedure and a fixed-point argument, and the key ingredient is the well known Sch…

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…

Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a $(3-p)$-form, and a $4$-form. This allows the introduction of techniques from parabolic…

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their…

Using the $L^2$-norm of the Higgs field as a Morse function, we count the number of connected components of the moduli space of parabolic $U(p,q)$-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. This space is homeomorphic to the…

Study Blaschke's asymptotic lines on surfaces in 3D space.

We study vector fields generating a local flow by automorphisms of a parabolic geometry with higher order fixed points. We develop general tools extending the techniques of [1], [2], and [3]. We apply these tools to almost Grassmannian, almost quaternionic, and contact parabolic geometries, including CR structures, to …

Proves existence of flat connection on theta functions for G-bundles.

Study of light function singularities on surfaces.

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of…

We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $|1|$--graded geometries which can …

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller compon…

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their …

In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket wi…

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence…

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord $AB$ on the parabola, let us denote by $P$ the point on the parabola where the tangent is parallel to $AB$ and by $V$ the point where the line through $P$ parallel to the axis of the p…

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear parabolic equations on compact Riemannian manifolds under the Ricci flow.

The paper solves a complex financial optimization problem using a novel mathematical technique.

When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related wit…

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum poly…