Latest in math.dg

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Homogeneous G-structuresJul 15 2019The theory of $G$-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional ... More
Integration by parts formula for non-pluripolar productJul 15 2019In this paper, we prove the integration by parts formula for the non-pluripolar product on a compact K\"ahler manifold. Our result generalizes the special case of potentials with small unbounded loci proved in [BEGZ10].
Décomposition solitonique des variétés toriquesJul 15 2019In this paper, we determine the solitonic decomposition of a Fano toric manifold by computing eigenfunctions of solitonic complex Laplacian operator.
Invariant PDEs on homogeneous manifolds via the affine structure of the bundles of jet spacesJul 14 2019Let $M$ be an $(n+1)$-dimensional manifold, let a group $G$ act transitively on $M$ and let $J^k(n,M)$ denote the space of $k$-jets of hypersurfaces of $M$. We make the following two assumptions on the action of $G$. First, there exists a hypersurface ... More
A KK-theoretic perspective on deformed Dirac operatorsJul 14 2019We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action ... More
Harmonic flow of geometric structuresJul 13 2019We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by C. M. Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, ... More
Analytically stable Higgs bundles on some non-Kähler manifoldsJul 13 2019In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily K\"ahler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.
Connection Cochain of Abelian Extensions and Connection $1$-FormsJul 13 2019In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure ... More
Geodesic orbit Finsler space with $K\geq0$ and the (FP) conditionJul 13 2019In this paper, we study the interaction between the geodesic orbit (g.o.~in short) property and certain flag curvature conditions. A Finsler manifold is called g.o.~if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the ... More
Non-existence of orthogonal coordinates on the complex and quaternionic projective spacesJul 12 2019DeTurck and Yang have shown that in the neighbourhood of every point of a $3$-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this property does ... More
A variational approach to the Hermitian-Einstein metrics and the Quot-scheme limit of Fubini-Study metricsJul 12 2019This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that ... More
Curvature-dimension conditions for diffusions under time changeJul 12 2019We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, ... More
Deformations of Vector Bundles over Lie GroupoidsJul 12 2019VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of ... More
Fibered Cusp b-Pseudodifferential Operators and its ApplicationsJul 12 2019Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle $\phi:\partial_0 X \to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus ... More
Nearly Frobenius AlgebrasJul 11 2019In this introductory paper we study nearly Frobenius algebras which are generalizations of the concept of a Frobenius algebra which appear naturally in topology: nearly Frobenius algebras have no traces (co-units). We survey the most basic foundational ... More
Optimal destabilization of K-unstable Fano varieties via stability thresholdsJul 11 2019We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano ... More
Some criteria for uniform K-stabilityJul 11 2019We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $\beta$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and ... More
Partial coherent state transforms, $G \times T$-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groupsJul 11 2019In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $G\times T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian ... More
Conformally related Douglas metrics are RandersJul 11 2019We show that two-dimensional conformally related Douglas metrics are Randers
Free boundary minimal surfaces in the unit ball : recent advances and open questionsJul 11 2019In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov ... More
Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroidsJul 11 2019In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on ... More
New classes of null hypersurfaces in indefinite Sasakian space-formsJul 10 2019We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overline{\phi},\zeta, \eta)$, tangent to the characteristic vector field $\zeta$, called; {\it contact screen conformal} and {\it contact screen umbilic} ... More
The dual volume of quasi-Fuchsian manifolds and the Weil-Petersson distanceJul 10 2019Making use of the dual Bonahon-Schl\"afli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson distance between the ... More
Symmetry defects and orbifolds of two-dimensional Yang-Mills theoryJul 10 2019We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to outer automorphisms of $G$, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path ... More
Killing forms on $2$-step nilmanifoldsJul 10 2019We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the ... More
Principal configurations around umbilics of spacelike surfaces in null hypersurfaces of $\mathbb{R}_1^4$Jul 10 2019We study the principal configurations around an isolated $\eta$-umbilical point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of Minkowski space $\mathbb{R}_1^4$ relative to a well-defined null vector field $\eta$ orthogonal to ... More
A two-piece property for free boundary minimal surfaces in the ballJul 09 2019We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean 3-ball into exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains ... More
Some analytic results on interpolating sesqui-harmonic mapsJul 09 2019In this article we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between ... More
On $S$-Curvature of Homogeneous Finsler spaces with $(α, β)$-metricsJul 09 2019The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with ... More
Constraints on families of smooth 4-manifolds from Bauer-Furuta invariantsJul 09 2019We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's ... More
Convex ancient solutions to mean curvature flowJul 09 2019X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation ... More
Real hypersurfaces in the complex quadric with Reeb parallel structure Jacobi operatorJul 09 2019In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_{\xi}$ and its derivative ... More
Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de RicciJul 09 2019Along the Ricci flow, we study the polyhomogeneity of complete Riemannian metrics endowed with "a Lie structure fibred at infinity", that is, a class of Lie structures at infinity that induce in a precise way a fibre bundle structure on a certain compactification ... More
The dihedral rigidity conjecture for n-cubesJul 08 2019We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional cube has nonnegative scalar curvature and weakly mean convex faces, ... More
Classification of generalized Kähler-Ricci solitons on complex surfacesJul 08 2019Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized K\"ahler in two distinct ways, with vanishing ... More
In search of stable geometric structuresJul 08 2019We will look for stable structures in four situations and discuss what is known and unknown.
Killing-Yano 2-forms on 2-step nilpotent Lie groupsJul 08 2019In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that ... More
Upper bounds for higher-order Poincar'e constantJul 08 2019We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply uniform upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of ... More
Sphere theorems for RCD and stratified spacesJul 08 2019Jul 09 2019We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.
An area bound for surfaces in Riemannian manifoldsJul 08 2019Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a multiple of ... More
Constant Q-curvature metrics on conic 4-manifoldsJul 08 2019We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.
Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel structure Jacobi operatorJul 08 2019We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, ... More
A flat torus theorem for convex co-compact actions of projective linear groupsJul 07 2019In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ ... More
Eigenvalue estimates via Hömander's $L^2$-methodJul 07 2019Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev ... More
Locally CAT(0)-metric is incompatible with PSC-metricJul 06 2019We generalize the concept of enlargeable to closed topology manifolds and then show that the connected sum of a closed manifold $M^n (n \leq 8)$ which admits locally CAT(0)-metric with an arbitrarily equal-dimensional closed manifold carries no Riemannian ... More
Solvable Lie algebras of vector fields and a Lie's conjectureJul 05 2019We present a local constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable ... More
A note about scalar curvature on the total space of a vector bundleJul 05 2019In this note, we construct complete Riemannian metrics to show that $M^n \times R^2$ admits complete Riemannian metrics with positive scalar curvature (PSC-metric) and that the total space of tangent bundles of orientable surfaces (except torus) admits ... More
An ambient approach to conformal geodesicsJul 05 2019Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is ... More
An ambient approach to conformal geodesicsJul 05 2019Jul 11 2019Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is ... More
An interior fixed point property of a convex domain in a Riemannian manifold with a poleJul 05 2019The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous bijection on the boundary of a bounded ... More
Non-diagonal invariant Einstein metrics on real flag manifoldsJul 05 2019We find the Einstein invariant metrics on the real flag manifolds $SO(4)/S(O(2)\times O(1)\times O(1)))$ and $(SO(l)\times SO(l))/S(O(l-1)\times O(1)),$ $l\geq 4$ and classify them by isometry. These are the only real flag manifolds of classical type ... More
On the topology and the boundary of N-dimensional RCD(K,N) spacesJul 04 2019We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, ... More
Two-dimensional twistor manifolds and Teukolsky operatorsJul 04 2019The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection ... More
Degree theory for orbifoldsJul 04 2019In [3] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values ... More
Openness of K-semistability for Fano varietiesJul 04 2019In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place ... More
Harmonic almost contact metric manifolds revisitedJul 04 2019The study of harmonicity for almost contact metric structures was initiated by Vergara-D\'iaz and Wood and continued by Gonz\'alez-D\'avila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric ... More
The Light Ray transform on Lorentzian manifoldsJul 04 2019We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities ... More
Second order rectifiability of varifolds of bounded mean curvatureJul 03 2019We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$ dimensional submanifolds ... More
Incomplete Yamabe flows and removable singularitiesJul 03 2019We show that any Yamabe flow starting from any Riemannian manifold of dimension $m\geq3$ minus a closed submanifold of dimension $n<\frac{m-2}{2}$ must remain geodesically incomplete as long as the flow exists. This is contrasted with the two-dimensional ... More
Bartnik mass via vacuum extensionsJul 03 2019We construct asymptotically flat, scalar flat extensions of Bartnik data $(\Sigma, \gamma, H)$, where $\gamma$ is a metric of positive Gauss curvature on a two-sphere $\Sigma$, and $H$ is a function that is either positive or identically zero on $\Sigma$, ... More
Spacetime positive mass theorems for initial data sets with noncompact boundaryJul 03 2019In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a noncompact boundary. Under suitable dominant energy conditions imposed both on the interior ... More
Spacetime positive mass theorems for initial data sets with noncompact boundaryJul 03 2019Jul 08 2019In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a noncompact boundary. Under suitable dominant energy conditions imposed both on the interior ... More
Holonomy groups of compact flat solvmanifoldsJul 03 2019This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is ... More
Anti-invariant Riemannian submersions from locally conformal Kaehler manifoldsJul 03 2019B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant ... More
Curves in a spacelike hypersurface in the Minkowski space-timeJul 03 2019Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski ... More
Local rigidity of manifolds with hyperbolic cusps I. Linear theory and pseudodifferential calculusJul 03 2019This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre (arXiv:1806.04218) on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to ... More
Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski spaceJul 03 2019Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved ... More
Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski spaceJul 03 2019Jul 04 2019Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved ... More
Time analyticity for the heat equation and Navier-Stokes equationsJul 03 2019We prove the analyticity in time for non-decaying solutions of two parabolic equations in the whole space. One of them involves solutions to the heat equation of double exponential growth on $\M$. Here $\M$ is $\R^n$ or a complete noncompact manifold ... More
On Pseudo-Umbilical Spacelike Submanifolds in Indefinite Space Form Mn+p p (c)Jul 02 2019In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result ... More
Optimal lower bounds for Donaldson's $\mathcal{J}$-functionalJul 02 2019In this paper we study optimal lower bounds for Donaldson's J-functional, and give an explicit formula for precisely how far it is from being proper. As a main application this leads to new sufficient conditions for existence of constant scalar curvature ... More
Rational curves on lattice-polarised K3 surfacesJul 02 2019Fix a K3 lattice $\Lambda$ of rank two and $L\in\Lambda$ a big and nef divisor that is positive enough. We prove that the generic $\Lambda$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening ... More
Curves on K3 surfacesJul 02 2019We show that every projective K3 surface over an algebraically closed field of characteristic zero contains infinitely many rational curves. For this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. Regeneration, a ... More
Nonlinear spectrums of Finsler manifoldsJul 02 2019In this paper, we study the spectral problem in Finsler geometry. The spectrum of a Finsler metric measure manifold is defined to be the set of the critical values of the canonical energy functional, which is captured by a faithful dimension-like function. ... More
Nonlinear stability of explicit self-similar solutions for the timelike extremal hypersurfaces in R^{1+3}Jul 02 2019This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$. We find there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal ... More
Minimizing Valuation is Quasi-monomialJul 02 2019We prove a version of Jonsson-Musta\c{t}\v{a}'s Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of ... More
Equivariant instanton homologyJul 01 2019We define four versions of equivariant instanton Floer homology ($I^+, I^-, I^\infty$ and $\widetilde I$) for a class of 3-manifolds and $SO(3)$-bundles over them including all rational homology spheres. These versions are analogous to the four flavors ... More
Optimization on flag manifoldsJul 01 2019A flag is a sequence of nested subspace. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise as Krylov subspaces in matrix computations, and as multiresolution ... More
The CR Ahlfors derivative and a new invariant for spherically equivalent CR mapsJul 01 2019We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe [23] that generalizes the CR Schwarzian derivative studied earlier by the second-named author [21]. This notion possesses several important properties similar to those ... More
Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-spaceJul 01 2019Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\boldsymbol R^3_1$. A complete light-like line in $\boldsymbol R^3_1$ is called an entire null line on the surface $S$ in $\boldsymbol R^3_1$ if it lies on $S$ and consists of only null ... More
The ring of normal densities, Gelfand transform and smooth BKK-type theoremsJul 01 2019We suggest an algorithm allowing to obtain some new integral-geometric formulae from the existing formulae of Crofton type. These new formulae are applied to get smooth versions of BKK theorem. The algorithm is based on the calculations in the ring of ... More
Linear Independence of Covariant Derivatives and Space-CurvaturesJul 01 2019It is developed the considerations from (S. M. Min\v{c}i\'c, [14, 15]) about curvature tensors and pseudotensors for a non-symmetric affine connection space in this paper. How many kinds of covariant derivatives are enough to be defined for complete researching ... More
Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time SystemsJul 01 2019We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For flat continuous-time systems, no comparable result is available. The advantage ... More
On Two notions of Gerbe over StackJun 30 2019Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles define a geometric stack. On the other hand, given a geometric stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ ... More
Causal geodesic incompleteness of spacetimes arising from IMP gluingJun 29 2019In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. ... More
Operational total space theory of principal 2-bundles II: 2-connections and 1- and 2--gauge transformationsJun 29 2019The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the contraction and ... More
Neck-Pinching of $CP^1$-structures in the PSL(2,C)-character varietyJun 28 2019Jul 02 2019Let S be a closed oriented surface of genus at least two. We consider a path of $CP^1$-structures $C_t$ on S leaving every compact subset in the deformation space of (marked) $CP^1$-structures on S, such that its holonomy converges in the PSL(2, C)-character ... More
Engel structures on complex surfacesJun 28 2019We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting ... More
A neighbourhood theorem for submanifolds in generalized complex geometryJun 28 2019We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic ... More
On super Plücker embedding and possible application to cluster algebrasJun 28 2019We define a super analog of the classical Pl\"{u}cker embedding of the Grassmannian into a projective space. Only a very special case was considered before in the literature. The "super Pl\"{u}cker map" that we introduce takes the Grassmann supermanifold ... More
Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flowJun 27 2019We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we ... More
On Warped Product Gradient Ricci-Harmonic SolitonJun 27 2019In this paper we study gradient Ricci-Harmonic soliton with structure of warped product manifold. We obtain some triviality results for the potential function, warping function and the harmonic map which reaches maximum or minimum. In order to obtain ... More
Positive harmonic functions on covering spacesJun 27 2019We show that if $p \colon M \to N$ is a normal Riemannian covering, with $N$ closed, and $M$ has exponential volume growth, then there are non-constant, positive harmonic functions on $M$. This was conjectured by Lyons and Sullivan in \cite{LS}.
Equivariant discretizations of diffusions, random walks, and harmonic functionsJun 27 2019For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.
Visual Curve Completion and Rotational Surfaces of Constant Negative CurvatureJun 27 2019If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves ... More
Hermitian curvature flow on locally homogeneous complex surfacesJun 27 2019We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behaviour of the solutions to the flow. Finally, we compute the Gromov-Hausdorff limit of immortal ... More
Einstein Metrics, Projective Structures and the $SU(\infty)$ Toda EquationJun 27 2019We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, ... More
A step to Gronwall's conjectureJun 27 2019In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit ... More
Stability of the Spacetime Positive Mass Theorem in Spherical SymmetryJun 26 2019The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In ... More
Geometrically finite Poincaré-Einstein metricsJun 26 2019We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted H\"older spaces.