arXiv research

Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.

In this article, we discuss which semisimple locally symmetric spaces admit an AHS--structure invariant to local symmetries. We classify them for all types of AHS--structures and determine possible equivalence classes of such AHS--structures.

We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds.

An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetri…

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…

We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsi…

In this paper the notion of an M-th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneo…

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time e…

We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit description. This allows us to describe the structure of symmetric geometric structur…

An differential field $(F;\partial_1,...,\partial_m)$ of characteristic zero, a subgroup $H$ of affine group $GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, \partial >^H=\{f^{\part…

The $L^2$-$\partial\overline\partial$-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.

Sharp Hölder regularity found for complex Frobenius theorem coordinates.

In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…

Two-dimensional Riemannian manifolds uniquely determined by boundary data.

Study cohomologies of complex manifolds with symplectic forms and their stability.

Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\partial_i$ be the component of $\partial M_i$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$ and $M^*$, where $M^i=M…

We present a simple, flexible, and general framework titled Partial Registration Network (PRNet), for partial-to-partial point cloud registration. Inspired by recently-proposed learning-based methods for registration, we use deep networks to tackle non-convexity of the alignment and partial correspondence problems. Whi…

The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.

The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset Π_1$, $\partial D_2 \subset Π_2$, $Π_1$ and $Π_2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$). Suppose that $M$ is contained in the slab determined by these hyperplanes and that the mean cu…

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

Paper tackles distribution matching by partially matching distributions, achieving robust results.

We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of…

Differential structure on partial isometries over Grassmannian constructed.

Ribbon cobordism forms a partial order in 3-manifolds.

On a compact complex manifold $X$, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if $X$ satisfies the $\partial\overline{\partial}$-Lemma.

Abstract study of HKT manifolds, proving Hodge theory and formality properties.

On a compact $\partial\bar\partial$-manifold $X$, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as $H_{dR}^k (X)=\oplus_{p+q=k}H^{p,\,q}(X)$, where the $H^{p,\,q}(X)$ are canonically isomorphic to the Dolbeault cohomology groups $H_{\bar\partial}^{p,\,q}(X)$. F…

Study on $4$-dimensional almost-Hermitian manifolds, proving $\overline\partial$-harmonic forms invariant under certain metrics.

We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between partial matchings. We introduce the notion of the lattice presentation of a par…

It is shown that any smooth strictly convex global solution of $\det(\frac{\partial^{2}u}{\partial ξ_{i}\partial ξ_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial ξ_{i}} - d_0\right\},$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

3-manifold curvature comparison with rotationally symmetric bodies.

Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.

We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$. Although $\partial_\infty L_2…

Paper categorifies a polynomial related to ribbon graphs.

New algorithm improves reinforcement learning from partial observations.

Ribbon cobordisms form a partial order on 3-manifolds.

New findings on complex manifold properties under deformations.

Anosov flow found in specific partially hyperbolic systems.

For any atoroidal iwip $φ\in Out(F_N)$ the mapping torus group $G_φ=F_N\rtimes_φ<t>e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_φ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_φ$. We prove that for any $φ$ as above…

We solve the following problem for $n=2:$ Is any n-dimensional Finsler manifold $(M, F)$ with a function $f$ which is nonconstant and smooth on $M$ satisfying $\dfrac{\partial g^{ij}}{\partial y^k}\dfrac{\partial f}{\partial x^i}=0,$ a Riemannian manifold? The problem for $n>2$ remains open.

New complex non-Kähler manifolds with specific properties are constructed.

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group $G$ acts geometrically (i.e. properly and cocompactly by isometries) on a CAT(0) space $X$. (Such group $G$ is called a {\it CAT(0) group}.) Then the group $G$ acts by homeomorphisms on the boundary $\part…

A new boundary for geodesic spaces defined and studied.

New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.