arXiv research

The article shows how to count small eigenvalues without assuming Morse functions.

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code. PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan, et al. We present the set of PD-codes as a stand alone object and discuss its relationship with link diagrams. We give an explicit …

New algorithm reduces dynamic regret by adapting to comparator complexity.

Let $l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\| l_1-\bar l_0\| \ge \| l_1-l_0\|, \| \bar l_1-l_0\| \ge \| \bar l_1-\bar l_0\|.$ He…

Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\bar p}H_*(X\times Y)$ to $I^{\bar p}H_*(X)$ and $I^{\bar p}H_*(Y)$, provided that the perversity $\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem r…

The objective of the present paper is to study the $η$-Ricci solitons on Kenmotsu manifold with generalized symmetric metric connection of type $(α,β)$. There are discussed Ricci and $η$-Ricci solitons with generalized symmetric metric connection of type $(α,β)$ satisfying the conditions $\bar{R}.\bar{S}=0$, $\bar{S}.\…

Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is well-defined. We study the short-time behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\b…

Let $( \bar g, \bar k)$ be a solution to the maximal constraint equations of general relativity on the unit ball $B_1$ of $\mathbb{R}^3$. We prove that if $(\bar g,\bar k)$ is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution $(g,k)$ on $\mathbb{R}^3$ that ext…

Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $\bar S$, which control non-associativity of the representation product in a peculiar channel $R\otimes \bar R \otimes R \longrightarrow R$. These $\bar S$ are simultaneously symmetric and orthogo…

Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci tensor and the section curvature of $(\bar{M},\bar{g})$ is nonpositive. Moreover, other …

Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure theorems for $M$; in particular we show that there is no loss of generality in assumi…

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that $\bullet$ $\mathcal{C}_a$ is an unstable minimal surface with index o…

Let $(M^n,g_0)$ and $(\bar{M}^{n+1},\bar{g})$ be complete Riemannian manifolds with $|\bar{\nabla}^k\bar{Rm}|\le \bar{C}$ for $k \le 2$, and suppose there is an isometric immersion $F_0: M^n \rightarrow \bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \rightarrow \bar{M}^{n+1}$ ($t\in [0,T]$) be a fam…

In this article we extend Milnor's fibration theorem for complex singularities to the case of singularities $f \bar g:(X,P) \to (C,0))$ defined on a complex analytic singularity germ $(X,P)$, with $f, g$ holomorphic and $f \bar g$ having an isolated critical value at $0 \in C$. This can also be regarded as a result for…

In this article, we prove that the equation of the Schrödinger maps from ${\bf R}^2$ to the hyperbolic 2-space ${\bf H}^2$ is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schrödinger-type system of unknown three complex functions $p,q,r$ and a real function $u$: {c} iq_t+q_{z{\bar z}}-2u q+2({\ba…

Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on th…

Spatial refinement of Bar-Natan homology constructed.

We extend the notion of the symmetric signature $σ(\bar{M},r)$ in L^n(R) for a compact n-dimensional manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the …

In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact Kähler manifolds as well as a short time existence of the negative …

We construct a general procedure to extract the exclusive Racah matrices S and \bar S from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R =[1], [2], [3] and [2,2]. The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrig…

We prove, under a certain boundedness condition at infinity on the $(\bar{X}^{\top}, \bar{X}^{\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\bar{X}$-bounded and $\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly con…

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar Q(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar Q(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar Q(3,3)$ is isomorphic to…

In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. Considering a relative normalization $\bar{y}$ of an hypersurface $Φ$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar…

This paper is devoted to the systematic investigation of the cone construction for Riemannian $G$ manifolds M, endowed with an invariant metric connection with skew torsion $\nabla^c$, a `characteristic connection'. We show how to define a $\bar G$ structure on the cone $\bar M=M\x \R^+$ with a cone metric, and we prov…

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $Ω$ with $\partial Ω$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $Ω$ and the mean curvature $…

We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset\mathcal{M}$ and a continuous map $X:\bar{M}\to\mathbb{C}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain…

Lower bounds on ribbon distance using Bar-Natan and α-Homology.

Let $f$ be a smooth plurisubharmonic function which solves $$ \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }Ω\subset \mathbb C^n.$$ Suppose that the metric $ω_{f}=\sqrt{-1}f_{i\bar j}dz_{i}\wedge d\bar z_{j}$ is complete and $f$ satisfies the growth condition $$ C^{-1}(1+|z|^2)\leq f\leq C(1+ |z|^2),\;\;\;\; as\;\;\; |z|\to…

Suppose $S$ is a surface of genus $\ge 2$, $f: S \to S$ is a surface homeomorphism isotopic to a pseudo-Anosov map $α$ and suppose $\ti S$ is the universal cover of $S$ and $F$ and $A$ are lifts of $f$ and $α$ respectively. We show there is a semiconjugacy $Θ: \ti S \to \bar Ł^s \times \bar Ł^u$ from $F$ to $\bar A$, …

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this …

Generalisation of the El Farol bar problem to that of many bars here leads to the Kolkata restaurant problem, where the decision to go to any restaurant or not is much simpler (depending on the previous experience of course, as in the El Farol bar problem). This generalised problem can be exactly analysed in some limit…

For a smooth strictly plurisubharmonic function $u$ on a open set $Ω\subset\mathbb{C}^{n}$ and $F$ a $C^{1}$ nondecreasing function on $\mathbf{R}^{*}_{+}$, we investigate the complex partial differential equations $Δ_{g}\log\det(u_{i\bar j})=F(\det(u_{i\bar j}))\Vert\nabla_{g}\log\det(u_{i\bar j})\Vert_{g}^{2},$ whe…

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.

Holographic principle matches deformed Liouville theory action.

We study Einstein's equation in $(m+n)D$ and $(1+n)D$ warped spaces $(\bar{M},\bar{g})$ and classify all such spaces satisfying Einstein equations $\bar{G}=-\barΛ\bar{g}$. We show that the warping function not only can determine the cosmological constant $\barΛ$ but also it can determine the cosmological constant $Λ$ a…

We show that the order of torsion homology classes in Bar-Natan deformation of Khovanov homology is a lower bound for the unknotting number. We give examples of knots that this is a better lower bound than |s(K)/2|, where s(K) is the Rasmussen s invariant defined by the Bar-Natan spectral sequence.

We construct smooth 4-manifolds homeomorphic but not diffeomorphic to CP^2+6CP^2-bar.

We classify the smallest finite volume complex hyperbolic surfaces with cusps which admit smooth toroidal compactifications and which are not birational to a bi-elliptic surface. Remarkably, there is only one such surface which appears to be the compactification of a Picard modular surface.

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then t…

The paper examines compactifications of Poincaré-Einstein manifolds and their convergence properties.

This paper extends T-duality to exotic chiral de Rham complexes.

In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, ε}$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and i…

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,φ_{\bar{D}})$ where $φ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D…

Let $M$ be an $n-$dimensional differentiable manifold equipped with a torsion-free linear connection $\nabla$ and $T^{\ast }M$ its cotangent bundle. The present paper aims to study a metric connection $\widetilde{% \nabla }$ with nonvanishing torsion on $T^{\ast }M$ with modified Riemannian extension ${}\bar{g}_{\nabl…

New algebraic framework for studying surfaces in 3-manifolds.

Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like current…