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Scalar Flat Compactifications Of Poincar{é}-einstein Manifolds And ApplicationsSep 18 2019We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{\'e}-Einstein manifolds. As a first consequence , we obtain a sharp lower bound for the first ... More

Quasiregular curvesSep 18 2019We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. ... More

Ricci Solitons, Conical Singularities, and NonuniquenessSep 17 2019In dimension $n=3$, there is a complete theory of weak solutions of Ricci flow - the singular Ricci flows introduced by Kleiner and Lott - which are unique across singularities, as was proved by Bamler and Kleiner. We show that uniqueness should not be ... More

Infinitesimal symmetries in Contact Hamiltonian systemsSep 17 2019In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, ... More

Field line winding of braided vector fields in tubular subdomainsSep 17 2019Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's ... More

Markov Processes with Jumps on Manifolds and Lie GroupsSep 17 2019We review some developments concerning Markov and Feller processes with jumps in geometric settings. These include stochastic differential equations in Markus canonical form, the Courr\`{e}ge theorem on Lie groups, and invariant Markov processes on manifolds ... More

A Correspondence Between Maximal Surfaces and Timelike Minimal Surfaces in $\mathbb{L}^3$Sep 17 2019We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space $\mathbb{L}^3$ that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in $\mathbb{L}^3$. There ... More

$\mathrm{RCD}(K,N)$ spaces and the geometry of multi-particle Schrödinger semigroupsSep 17 2019Sep 18 2019With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha\in [0,1]$ we introduce the $\alpha$-Kato class ... More

$\mathrm{RCD}(K,N)$ spaces and the geometry of multi-particle Schrödinger semigroupsSep 17 2019With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha\in [0,1]$ we introduce the $\alpha$-Kato class ... More

Geometric and spectral properties of directed graphs under a lower Ricci curvature boundSep 17 2019For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. ... More

Natural maps for measurable cocycles of compact hyperbolic manifoldsSep 17 2019Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra ... More

Momentum Ray Transforms, II: Range Characterization In the Schwartz spaceSep 17 2019The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,\xi)=\int_{-\infty}^\infty t^k\l f(x+t\xi),\xi^m\r\,dt. $ We give the range characterization for the operator ... More

Asymptotic expansions of solutions of the Yamabe equation and the $σ_k$-Yamabe equation near isolated singular pointsSep 16 2019We study asymptotic behaviors of positive solutions to the Yamabe equation and the $\sigma$k-Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, ... More

The spacelike-characteristic Cauchy problem of general relativity in low regularitySep 16 2019In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and the outgoing ... More

The canonical foliation on null hypersurfaces in low regularitySep 16 2019Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by Klainerman ... More

Topology of weak $G$-bundles via Coulomb gauges in critical dimensionsSep 16 2019The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can ... More

Relating second order geometry of manifolds through projections and normal sectionsSep 16 2019We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example we show how ... More

Quadratic differentials and circle patterns on complex projective toriSep 16 2019Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a ... More

Exotic Courant algebroids and T-dualitySep 16 2019In this paper, we extend the T-duality isomorphism by Gualtieri and Cavalcanti, from invariant exact Courant algebroids, to exotic exact Courant algebroids such that the momentum and winding numbers are exchanged, filling in a gap in the literature.

Pluripotential solutions versus viscosity solutions to complex Monge-Amp{è}re flowsSep 16 2019We compare various notions of weak subsolutions to degenerate complex Monge-Amp{\`e}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong ... More

Weak forms of $\ddb-$Lemma on compact complex manifoldsSep 16 2019This paper is devoted to give a complete unified study of several weak forms of $\ddb-$Lemma on compact complex manifolds.

The Minkowski inequality in de Sitter spaceSep 15 2019The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained inverse mean ... More

The geometric quantizations and the measured Gromov-Hausdorff convergencesSep 15 2019On a compact symplectic manifold $(X,\omega)$ with a prequantum line bundle $(L,\nabla,h)$, we consider the one-parameter family of $\omega$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. ... More

Fundamental domains in ${\rm PSL}(2,{\mathbb R})$ for Fuchsian groupsSep 15 2019In this paper, we provide a necessary and sufficient condition for a set in ${\rm PSL}(2,{\mathbb R})$ or in $T^1{\mathbb H}^2$ to be a fundamental domain of a given Fuchsian group via its respective fundamental domain in the hyperbolic plane ${\mathbb ... More

Critical Clearing Time Sensitivity for Inequality Constrained SystemsSep 15 2019With the growth of renewable generation (RG) and the development of associated ride through curves serving as operating limits, during disturbances, on violation of these limits, the power system is at risk of losing large amounts of generation. In order ... More

Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone anglesSep 14 2019For deformation of 3-dimensional hyperbolic cone structures about cone angles $\theta$, the local rigidity is known for $0 \leq \theta < 2\pi$, but the global rigidity is known only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima ... More

Harmonic Gauss maps of submanifolds of arbitrary codimension and applicationsSep 14 2019It is extended to arbitrary codimension several results about the harmonicity of a Gauss map in the Euclidean space and in certain homogeneous spaces. It is introduced the concept of a harmonic unit normal section and proved that if a submanifold of $\mathbb ... More

Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical EndSep 14 2019If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable ... More

Non-holonomic equations for the normal extremals in geometric control theorySep 14 2019We provide a new and simple system of equations for the normal sub-Riemannian geodesics. These use a partial connection that we show is canonically available, given a choice of complement to the distribution. We also describe conditions which, if satisfied, ... More

A shape optimizaion problem for the first mixed Steklov-Dirichlet eigenvalueSep 14 2019We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem

Differential operators on almost-Hermitian manifolds and harmonic formsSep 14 2019We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic ... More

Yamabe Solitons on (LCS)$_n$-manifoldsSep 14 2019The object of the present paper is to study some properties of (LCS)$_n$-manifolds whose metric is Yamabe soliton. We establish some characterization of (LCS)$_n$-manifolds when the soliton becomes steady. Next we have studied some certain curvature conditions ... More

Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian ManifoldsSep 14 2019In this paper, we show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the ... More

Uniqueness and nonuniqueness of limits of Teichmueller harmonic map flowSep 13 2019The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, ... More

Degenerate elastic networksSep 13 2019We minimize a linear combination of the Willmore and the length functional among networks in $\mathbb{R}^d$ belonging to a given class determined by the number of curves, the order of the junctions and the angles between curves at the junctions. Since ... More

Steiner's formula and a variational proof of the isoperimetric inequalitySep 13 2019We give a proof of the isoperimetric inequality in the plane based on Steiner's formula for the area of a convex neighborhood.

Almost hypercomplex manifolds with Hermitian-Norden metrics and 4-dimensional indecomposable real Lie algebras depending on two parametersSep 13 2019The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional indecomposable real ... More

Volume entropy and lengths of homotopically independent loopsSep 13 2019This paper presents a new inequality for closed Riemannian manifolds involving the volume entropy and the set of lengths of any family of homotopically independent loops based at the same point. This inequality implies a curvature free collar theorem, ... More

On the uniqueness of Schwarzschild-de Sitter spacetimeSep 12 2019We establish a new uniqueness theorem for the three dimensional Schwarzschild-de Sitter metrics. For this some new or improved tools are developed. These include a reverse Lojasiewicz inequality, which holds in a neighborhood of the extremal points of ... More

The Adjunction Inequality for Weyl-Harmonic MapsSep 12 2019In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show that there is an Eells-Salamon type correspondence between nonvertical $\mathcal{J}$-holomorphic curves ... More

Equilibria of plane convex bodiesSep 12 2019We obtain a formula for the number of horizontal equilibria of a planar convex body $K$ with respect to a center of mass $O$ in terms of the winding number of the evolute of $\partial K$ with respect to $O$. The formula extends to the case where $O$ lies ... More

Warps and duality for double vector bundlesSep 12 2019A linear section of a double vector bundle is a parallel pair of sections which form a vector bundle morphism; examples include the complete lifts of vector fields to tangent bundles and the horizontal lifts arising from a connection in a vector bundle. ... More

Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvatureSep 12 2019In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under ... More

Infinitely many sign-changing solutions of a critical fractional equationSep 12 2019In this paper we prove the existence of an unbounded sequence of sign changing solutions to a laplacian fractional and critical problem in the Euclidean space by reducing the initial problem to an equivalent problem on the Euclidean unit sphere and exploring ... More

Harmonic Forms, Price Inequalities, and Benjamini-Schramm ConvergenceSep 12 2019We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature ... More

Quasi-morphisms on surface diffeomorphism groupsSep 12 2019We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm ... More

$\mathcal{Z}$-symmetries of $(ε)$-para-Sasakian 3-manifoldsSep 12 2019The object of this paper is study $(\epsilon)$-para-Sasakian 3-manifolds satisfying certain conditions on the $\mathcal{Z}$ tensor. We characterize, $\mathcal{Z}$-symmetric; $\mathcal{Z}$-semisymmetric; $\mathcal{Z}$-pseudosymmetric; and projectively ... More

Global higher order estimates for collapsing Calabi-Yau metrics on elliptic K3 surfacesSep 12 2019We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

Conformal Positive Mass Theorems for Manifolds with ChargeSep 12 2019In this paper, we prove conformal positive mass theorems for asymptotically flat manifolds with charge. We apply conformal relations to show that if the conformal sum of scalar curvature is not less than the norm square of electric field, the sum of the ... More

Positive energy theorem for asymptotically anti-de Sitter spacetimes with distributional curvatureSep 12 2019We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

Prescribed non positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equationSep 11 2019We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non ... More

Identifying Berwald Finsler GeometriesSep 11 2019Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We find a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. ... More

Preserve one, preserve allSep 11 2019Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X \rightarrow X$ preserving at least one level set $d^{-1}(r)$, with ... More

The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spacesSep 11 2019In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space ... More

Conjugate points for systems of second-order ordinary differential equationsSep 11 2019We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both ... More

L-infinity bialgebroids and homotopy Poisson structures on supermanifoldsSep 11 2019We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that $(TM, T^*M)$ has a canonical structure of an $L_{\infty}$-bialgebroid. ... More

Convex Integration Theory without IntegrationSep 11 2019We replace the usual Convex Integration formula by a Corrugation Process and introduce the notion of Kuiper differential relations. This notion provides a natural framework for the construction of solutions with self-similarity properties. We consider ... More

The Cheeger constant of an asymptotically locally hyperbolic manifold and the Yamabe type of its conformal infinitySep 11 2019Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$\gamma$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant ... More

An Alexandrov theorem in Minkowski spacetimeSep 11 2019In this paper, we generalize a theorem {\`a} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-spaceSep 11 2019It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as ... More

Some recent work on biharmonic conformal mapsSep 10 2019This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and ... More

Jacobi relations on naturally reductive homogeneous spacesSep 10 2019Naturally reductive spaces in general can be seen as an adequate generalization of symmetric spaces. Nevertheless there are non-symmetric naturally reductive spaces whose geometric properties come closer to symmetric spaces than others. We consider a ... More

Wall crossing for K-moduli spaces of plane curvesSep 10 2019We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing ... More

Properties of the Null Distance and Spacetime ConvergenceSep 10 2019The notion of null distance for Lorentzian manifolds recently introduced by Sormani and Vega gives rise to a metric and induces the manifold topology under mild assumptions on the time function of the spacetime. We show that, endowed with a (locally) ... More

Properties of the Null Distance and Spacetime ConvergenceSep 10 2019Sep 16 2019The notion of null distance for Lorentzian manifolds recently introduced by Sormani and Vega gives rise to an intrinsic metric and induces the manifold topology under mild assumptions on the time function of the spacetime. We prove that warped products ... More

Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvatureSep 10 2019Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting ... More

Exactness of Lepage 2-forms and globally variational differential equationsSep 10 2019The exactness equation for Lepage 2-forms, associated with variational systems of ordinary differential equations on smooth manifolds, is analyzed with the aim to construct a concrete global variational principle. It is shown that locally variational ... More

The Chiral Anomaly of the Free Fermion in Functorial Field TheorySep 10 2019When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, ... More

On Mean curvature flow of Singular Riemannian foliations: Non compact casesSep 10 2019In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions, ... More

On Mean curvature flow of Singular Riemannian foliations: Non compact casesSep 10 2019Sep 17 2019In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions, ... More

A Bochner Formula on Path Space for the Ricci FlowSep 09 2019We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. ... More

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie groupSep 09 2019Let $G$ be a complex Lie group acting on a compact complex Hermitian manifold $M$ by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute ... More

Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde FormulaSep 09 2019Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum ... More

Orthogonal Higgs bundles with singular spectral curvesSep 09 2019We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral ... More

Orthogonal Higgs bundles with singular spectral curvesSep 09 2019Sep 10 2019We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral ... More

$S^1$-quotient of $Spin(7)$-structuresSep 09 2019If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a $G_2$-structure. We derive equations relating the intrinsic torsion of the $Spin(7)$-structure to that ... More

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifoldsSep 09 2019Given $(X,\omega)$ compact K\"ahler manifold and $\psi\in\mathcal{M}^{+}\subset PSH(X,\omega)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(\omega+dd^{c}\psi)^{n}>0$, we prove that the ... More

Exploration of Balanced Metrics on Symmetric Positive Definite MatricesSep 09 2019Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose ... More

Topologies on the future causal completionSep 09 2019Sep 14 2019We consider two topologies on the Geroch-Kronheimer-Penrose future completion of spacetimes, showing that their respective avantages are in principle mutually exclusive.

Topologies on the future causal completionSep 09 2019We consider two topologies on the Geroch-Kronheimer-Penrose future completion of spacetimes, showing that their respective avantages are in principle mutually exclusive.

Bergman kernel on Riemann surfaces and Kaehler metric on symmetric productsSep 09 2019Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincar\'e metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $\Omega^{\otimes k}_X$, where $\O$ is the holomorphic ... More

Uryson width and volumeSep 09 2019We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich-Lishak-Nabutovsky-Rotman.

Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequalitySep 09 2019By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by B\"or\"oczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, ... More

A class of curvature type equationsSep 09 2019In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of ... More

Deformations of Dolbeault cohomology classesSep 09 2019In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will paly the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, ... More

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebrasSep 08 2019Sep 10 2019The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study ... More

Bubble Tree Convergence of Conformally Cross Product Preserving MapsSep 08 2019We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith ... More

The Partial $C^{0}$-estimate along a general continuity path and applicationsSep 08 2019We establish a new partial $C^{0}$-estimate along a continuity path mixed with conic singularities along a simple normal crossing divisor and a positive twisted $(1,1)$-form on Fano manifolds. As an application, this estimate enables us to show the reductivity ... More

On the spectral characterization of Besse and Zoll Reeb flowsSep 07 2019A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent ... More

Ruh-Vilms Theorems For Minimal Surfaces Without Complex Points and Minimal Lagrangian Surfaces in $\mathbb C P^2$Sep 07 2019In this paper we investigate surfaces in $\mathbb C P^2$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion ... More

Handle attachment and the normalized first eigenvalueSep 06 2019We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximizing metrics for the normalized first eigenvalue ... More

Sharp asymptotics of the first eigenvalue on some degenerating surfacesSep 06 2019We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly ... More

Schwarz type lemmas for pseudo-Hermitian manifoldsSep 06 2019In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison ... More

Geometry of submanifolds with respect to ambient vector fieldsSep 06 2019Given a Riemannian manifold $N^n$ and ${\cal Z}\in \mathfrak{X}(N)$, an isometric immersion $f\colon M^m\to N^n$ is said to have the \emph{constant ratio property with respect to ${\cal Z}$} either if the tangent component ${\cal Z}^T_f$ of ${\cal Z}$ ... More

Aeppli cohomology and Gauduchon metricsSep 06 2019Let $(M,J,g,\omega)$ be a complete Hermitian manifold of complex dimension $n\ge2$. Let $1\le p\le n-1$ and assume that $\omega^{n-p}$ is $(\partial+\overline{\partial})$-bounded. We prove that, if $\psi$ is an $L^2$ and $d$-closed $(p,0)$-form on $M$, ... More

An atlas adapted to the Toda flowSep 05 2019We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to ... More

Finite Euclidean and Non-Euclidean GeometriesSep 05 2019The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After a brief introduction, ... More

A remark on the fundamental group of a compact negatively curved manifoldSep 05 2019In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong ... More

Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$Sep 05 2019Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at $-\infty$, generalizing a theorem for cylinders in [CM19b]. In the ... More

Complex Hessian equations with prescribed singularity on compact Kähler manifoldsSep 05 2019Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $\omega$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then ... More