Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.
problem Monotonicity of parabolic frequency on manifolds.
method Analyzes parabolic frequency function on manifolds, proving monotonicity without curvature assumptions.
result Monotonicity of parabolic frequency on all manifolds, no curvature assumption needed.
The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.
problem Backwards uniqueness for solutions of parabolic equations on Ricci flows.
method Defines and proves monotonicity of parabolic frequency for Ricci flow solutions.
result Backwards uniqueness for solutions of parabolic equations on Ricci flows.
The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.
This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The…
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
problem Gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
method Gradient estimates and Harnack inequalities for heat equations under the Laplacian G_2 flow.
result Monotonicity of parabolic frequency and backward uniqueness for positive solutions.
The paper improves heat equation estimates under weaker Ricci curvature conditions.
problem Improving heat equation estimates under weaker Ricci curvature conditions.
method Establishing Li-Yau-type and Hamilton-type estimates for positive solutions of the heat equation under generalized Ricci flow.
result Deriving Harnack-type inequalities and monotonicity of parabolic frequency.
New method proves heat flow of harmonic maps into CAT(0) spaces.
problem Global existence and uniqueness of suitable weak solutions.
method Elliptic regularization and variational structure.
result Lipschitz continuity of solutions in spatial variables.
For classification of the high frequency trading quantities, waiting times, price increments within and between sessions are referred to as the a-, b-, and c-increments. Statistics of the a-b-c-increments are computed for the Time & Sales records posted by the Chicago Mercantile Exchange Group for the futures traded on…
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
Criterion found for Lie algebroid connections on parabolic bundles.
problem Conditions for parabolic vector bundles to have Lie algebroid connections.
method Necessary and sufficient condition based on Lie algebroid structure.
result Found criterion for existence of parabolic Lie algebroid connections.
Computes deformations of parabolic structures on Riemann surfaces.
problem Infinitesimal deformations of parabolic connections and opers.
method Computes infinitesimal deformations of quadruples (X, S, E*, D) and (X, S, D).
result Monodromy map is an immersion from the moduli space of triples to the character variety.
Deep signature/log-signature FBSDE algorithm improves accuracy and training time.
problem Solving FBSDEs with state and path dependent features.
method Incorporates deep signature/log-signature transformation into RNN model.
result Improves accuracy and training time compared to existing methods.
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→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.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
Defines non-parabolic curves in spatial hybrid space with applications.
problem Defining and analyzing non-parabolic spatial hybrid framed curves.
method Definition and proof of existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
result Existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
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.
problem Nonuniformly parabolic boundary value problems with degeneration in normal direction.
method Theory of linear parabolic differential equations on noncompact Riemannian manifolds.
result Optimal solution theory for natural degeneration case.
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.
problem Understanding the structure of nodal and singular sets in parabolic inequalities.
method Establishing new estimates for the size and structure of nodal and singular sets using parabolic Lipschitz coefficients.
result Almost all nodal and singular sets are covered by regular parabolic Lipschitz graphs with estimates.
Sol3 is shown not to be parabolic.
problem Proving non-parabolicity of Sol3 method Analyzing left invariant metrics on Sol3 result Sol3 is non-parabolic Study gauge theory of real and quaternionic parabolic bundles over real curves.
problem Examining gauge theoretic aspects of real and quaternionic parabolic bundles over real curves.
method Investigate orbits of connections under gauge groups for fixed real or quaternionic structures.
result Gauge-theoretic quotients of real or quaternionic connections are inside the real points of moduli of holomorphic bundles.
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.
problem Defining and characterizing parabolic subgroups in complex braid groups.
method Introducing and studying parabolic subgroups of generalized braid groups associated with complex reflection groups.
result Parabolic subgroups form a lattice in most cases, with specific properties and conjectures about hyperbolicity.
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.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
New method to determine parabolic surfaces invariant under Killing fields.
problem Characterizing parabolic invariant surfaces.
method Clear and practical characterization method for invariant surfaces.
result A new way to determine parabolicity of invariant surfaces.
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.
problem Preserving parabolic subgroups during lattice deformations.
method Analysis of deformations into SU(n,1) and SO(n+1,1).
result Existence of 1-parameter families of parabolic-preserving deformations.
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.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. Parabolic mapping class acts on curve graphs of infinite type surfaces.
problem Understanding parabolic isometries on curve graphs of infinite type surfaces.
method Fine curve graph tools to prove existence of parabolic isometries.
result Existence of parabolic isometries on graphs of curves of infinite type surfaces.
Quotients of torus endomorphisms have parabolic orbifolds.
problem Understanding the structure of quotients of torus endomorphisms.
method Analyzing the properties of torus endomorphisms and their quotients.
result Every quotient of a torus endomorphism has a parabolic orbifold.
Paper generalizes Higgs bundle limits to parabolic setting.
problem Generalizing Higgs bundle limits to parabolic setting.
method Gauge theoretic construction of moduli space of parabolic Higgs bundles.
result Conformal limit always exists and defines holomorphic sections.
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 aκ1+bκ2=c or aH+bK=c, where $a,b,c\in \r$ and, as usual, κi are the principal curvatur…
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.
problem Proving nonexistence results for parabolic inequalities on Riemannian manifolds.
method Using a test function argument and weighted volume growth assumptions.
result Established Liouville-type theorems for (p,q)-Laplacian operator inequalities. Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.
problem Understanding the dynamics of parabolic automorphisms on hyperkahler manifolds.
method Analyzing the action of parabolic automorphisms on the second cohomology group and fibers of Lagrangian fibrations.
result Parabolic automorphisms preserving Lagrangian fibrations act ergodically on the 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.