arXiv research

Study the spectrum of Page's metric on complex projective spaces.

Classifies Einstein metrics on 4-manifolds with specific symmetry groups.

In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally we give information on how the Page space is related to some other metrics on the …

We prove that the quasi-Einstein metrics found by Lü, Page and Pope on $\mathbb{C}P^{1}$-bundles over Fano Kähler-Einstein bases are conformally Kähler and that the Kähler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb…

The paper explores spectral sequences of complex manifolds with special metrics.

The paper extends Hawking--Page solutions to various spacetimes with singularities.

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invarian…

Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous sol…

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal…

Study on stability of minimal submanifolds in specific Einstein manifolds.

We consider compact complex surfaces with Hermitian metrics which are Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at 1,2, or 3 points, and the isometry group of the metric must contain a 2-torus. Thus the Page metric on CP2#(-CP2) is almost the only metric of this type.

Alternative proof of Dynnikov's three-page index for torus links.

Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all h…

PAGE optimizes nonconvex problems with optimal convergence rates.

To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the sp…

We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This …

We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…

Let (M,h) be a compact 4-dimensional Einstein manifold, and suppose that h is Hermitian with respect to some complex structure J on M. Then either (M,J,h) is Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2…

Optimizing an interactive system against a predefined online metric is particularly challenging, when the metric is computed from user feedback such as clicks and payments. The key challenge is the counterfactual nature: in the case of Web search, any change to a component of the search engine may result in a different…

The paper refines the three-page index for links, proving a new bound and characterizing specific links.

Optimizes web page freshness with limited crawling frequencies.

We construct a series of finitely presented semigroups. The centers of these semigroups encode uniquely up to rigid ambient isotopy in 3-space all non-oriented spatial graphs. This encoding is obtained by using three-page embeddings of graphs into the product of the line with the cone on three points. By exploiting thr…

Extracts main content from web pages using neural sequence labeling.

Extends deformation theory to higher-page analogues of manifolds.

Improved linear upper bound for ribbonlength of knots.

PAGE is a simple gradient estimator for nonconvex optimization problems.

New invariant distinguishes Legendrian surfaces in 5-manifolds.

It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page--Bérard-Bergery's ansatz \cite{P78,B82}. We also …

The spectral sequence's $E_k$-page is a link invariant for $k\ge 3$.

We construct a contact 5-manifold supported by infinitely many distinct open books with the identity monodromy and pairwise exotic Stein pages (i.e. pages are pairwise homeomorphic but non-diffeomorphic Stein fillings of a fixed contact 3-manifold), moreover we describe a process of generating infinitely many such exam…

Optimization is commonly employed to determine the content of web pages, such as to maximize conversions on landing pages or click-through rates on search engine result pages. Often the layout of these pages can be decoupled into several separate decisions. For example, the composition of a landing page may involve dec…

Generative models predict page quality without training, useful for low-resource settings.

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one associates w…

Model predicts web page parallelism for improved browser performance and energy.

Scalable web crawling using noisy change-indicating signals.

Freya PAGE optimizes nonconvex optimization with heterogeneous, asynchronous workers.

TailedTS dataset benchmarks heavy-tailed time series forecasting and periodicity quantification.

We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book for $S^3,$ with relatively low genus. Thus we produce open books with low genus p…

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski-Demianski metrics one obtains a family of local toric Kahler-Einstein metrics. These can be used to construct local Sasaki-Einstein metrics in five dimensions which are generalisations of the Y^{p,q} manifolds. In fact, we find that…

A new bandit algorithm for web page item display.

Algorithm constructs Kirby diagrams for 4D open books.

In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structu…

A new infinite series of Einstein metrics is constructed explicitly on S^2 x S^3, and the non-trivial S^3-bundle over S^2, containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr black hole. In the special case, the metrics reduce to the homogeneous E…

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components…

New non-Kähler manifolds constructed with specific properties.

Study homotopy groups of open books and their pages, pages, and bindings.

Given any closed, connected, orientable $3$--manifold and integers $g\geq g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodr…