arXiv research

Self-affine arcs without inner weak separation are parabolic segments.

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. T…

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…

Defines connections on parabolic vector bundles for Lie algebroids.

Criterion found for Lie algebroid connections on parabolic bundles.

Computes deformations of parabolic structures on Riemann surfaces.

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth c…

Classifies holomorphic parabolic geometries on complex manifolds.

Alternative proof classifies Kleinian groups with two parabolics.

Defines non-parabolic curves in spatial hybrid space with applications.

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…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an a…

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

New estimates for nodal and singular sets of parabolic inequalities.

$Sol_3$ is shown not to be parabolic.

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

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

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…

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 defines parabolic subgroups for complex braid groups and proves they form a lattice.

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…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

New method to determine parabolic surfaces invariant under Killing fields.

Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous. This is the first global theorem on parabolic geometries.

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…

Study of parabolic-preserving deformations of hyperbolic lattices.

We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

The study of special metrics in various parabolic geometries.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

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

Quotients of torus endomorphisms have parabolic orbifolds.

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

Paper generalizes Higgs bundle limits to parabolic setting.

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…

Extends parabolic study to flat hyperkähler manifolds.

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

A non-elementary Möbius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of \cite{GW} to obtain an additional structure for the parameter space, which we term the {\sl two-parabolic space}. …

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

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Segmental structure is a common pattern in many types of sequences such as phrases in human languages. In this paper, we present a probabilistic model for sequences via their segmentations. The probability of a segmented sequence is calculated as the product of the probabilities of all its segments, where each segment …

In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…

Bayesian nonparametric method segments multi-sequence time series data.

Segmentation has been a major task in neuroimaging. A large number of automated methods have been developed for segmenting healthy and diseased brain tissues. In recent years, deep learning techniques have attracted a lot of attention as a result of their high accuracy in different segmentation problems. We present a n…