arXiv research

New method to determine parabolic surfaces invariant under Killing fields.

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

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the i…

A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form $aκ_1+bκ_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual, $κ_i$ are the principal curvatur…

Parabolic mapping class acts on curve graphs of infinite type surfaces.

Study geodesic flow on symmetric surfaces to determine parabolic type.

We study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $aκ_1+bκ_2=c$, where $κ_i$ are the principal curvatures, and $a,b$ and $c$ are constant.

Computes deformations of parabolic structures on Riemann surfaces.

Criterion found for Lie algebroid connections on parabolic bundles.

Defines connections on parabolic vector bundles for Lie algebroids.

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…

Study proves correspondence for special bundles on complex surfaces.

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed…

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…

Characterizes parabolic flute surfaces with specific parameters.

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 …

Real slices of parabolic opers on Riemann surfaces are studied.

In this note, we prove that the holonomy map from the set of equivalence classes of projective structures of parabolic type on non compact surfaces to the set of equivalence classes of parabolic representations of the fundamental group of the surface to P SL 2 (C) is a local biholomorphism.

Researchers prove twists can make surfaces parabolic even with large cuff lengths.

We determine explicitly the structure of the automorphism group of a parabolic Inoue surface. We also describe the quotients of the surface by typical cyclic subgroups of the automorphism group.

The goal of this paper is to classify parametrically parabolic submanifolds in any codimension. First, we describe the ones that are ruled and show that they are the only parabolic submanifolds that admit an isometric immersion as a hypersurface. Then, we classify the nonruled ones by two different means. In fact, we p…

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Gri…

We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.

Study parabolicity of Riemann surfaces via Fenchel-Nielsen parameters.

New examples of extremal Kähler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iter…

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

Proves conditions for minimal surfaces in complex hyperbolic space.

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a characterization of $δ$-parabolicity we prove that under additional conditions on $\mathbb{M}$,…

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

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

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

We establish parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of R^3 which are within a sub-logarithmic factor of the exterior of a cone. Unlike previous work showing that these two properties hold for minimal surfaces-with-boundary contained between two catenoids, we do no…

Non-ergodic geodesic flow on Cantor tree surfaces found.

In this paper, we prove the existence and uniqueness theorem for parabolic conical metrics on Riemann surfaces in the situation of generalized real angles, positive, zero and negative, by complex analysis, and give an example of this theorem to clarify concrete expressions of parabolic metrics on the two-sphere and gen…

Given a discrete subgroup $Γ$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $Σ$ of the (extended) hyperbolic plane by $Γ$. If $Γ$ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vecto…

We classify homothetical surfaces with constant mean curvature in hyperbolic space.

We study surfaces of constant mean curvature which are invariant by oneparameter group of either rotational isometries or parabolic isometries, immersed into the homogeneous manifold PSL2(R,tau). Also, we give some applications.

The algebra of differential invariants under $SA_3(\mathbb{R})$ of generic parabolic surfaces $S^2 \subset \mathbb{R}^3$ with nonvanishing Pocchiola $4^{\text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomi…

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…

Extends Milnor's criterion to biharmonic functions.

Study of light function singularities on surfaces.

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…

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…

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations…

New proof shows special surfaces have finite type.

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M…

We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…