The study introduces surfaces in quasi-Fuchsian manifolds and their asymptotic properties.
problem Understanding surfaces in quasi-Fuchsian manifolds and their asymptotic behavior.
method Introducing asymptotically Poincaré families of surfaces and showing they foliate ends by convex surfaces.
result Asymptotically Poincaré families determine projective structures at infinity and constant curvature surfaces.
The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.
problem Finding a lower bound for the first Neumann eigenvalue of surfaces with asymptotically flat ends.
method Integration of Bouchner's formula with contributions from A. Lichnerowicz, S. Brendle, R. Tsiamis, and Poincare's constant.
result Obtains a lower bound for the first Neumann eigenvalue of properly embedded surfaces.
Characterizes solvability of J-equation on Kähler surfaces with singularities.
problem Solvability of J-equation on Kähler surfaces with Poincaré type singularities.
method Two-parameter continuity path for J-equation, Kähler metrics with Poincaré type singularities.
result Existence of Poincaré type solutions implies boundedness of K-energy.
The paper constructs Poincaré-Einstein 4-manifolds with various cusps.
problem Constructing Poincaré-Einstein 4-manifolds with cusps.
method Constructing metrics on (0,∞)imesN and (0,∞)imesP. result Infinite families of Einstein metrics on (0,∞)imesN and (0,∞)imesP. We study the Seiberg-Witten equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric which is "asymptotic" to a Poincaré type metric at infinity. Then we compute a lower bound for the L2-norm of scalar curvature on these…
The Poincaré series for surfaces with boundary extends to the complex plane.
problem Counting geodesics on surfaces with boundaries.
method Analytic continuation of Poincaré series.
result Poincaré series extend meromorphically to the whole complex plane.
Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincaré disc: a point on the unit circle). We prove here that, if two minimal rays have a common asymptotic direction…
Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.
problem Computing Cheeger constants for conformally compact asymptotically constant mean curvature submanifolds.
method Analyzes conformally compact asymptotically constant mean curvature submanifolds in asymptotically hyperbolic spaces.
result Identifies conditions for Cheeger constant equality and vanishing mean curvature.
Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a con…
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
New method defines conformal geodesics using surfaces in higher-dimensional manifolds.
problem Defining conformal geodesics in a novel way.
method Using surfaces in higher-dimensional manifolds and tractor calculus.
result Conformal geodesics are critical points of renormalized area in higher-dimensional spaces.
New proof of surface group theorem for 2D Poincaré duality groups.
problem Characterizing groups with specific algebraic properties.
method Analyzing amenability and homological isoperimetric inequalities.
result Groups satisfying certain conditions are either amenable or have linear homological isoperimetric inequalities.
We study an asymptotic Dirichlet problem for Weyl structures on asymptotically hyperbolic manifolds. By the bulk-boundary correspondence, or more precisely by the Fefferman-Graham theorem on Poincaré metrics, this leads to a natural extension of the notion of Branson's Q-curvature to Weyl structures on even-dimension…
Introduces Epstein-Poincaré surfaces for G-oper, generalizing classical construction.
problem Generalizing classical Epstein-Poincaré surfaces to complex Lie groups.
method Introduces Epstein-Poincaré surfaces for G-oper, providing a criterion for Anosov holonomy.
result Provides a criterion for the holonomy of G-oper to be Δ-Anosov.
Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.
problem Understanding the cohomology structure of SU(2) character varieties of surfaces.
method Explicit description of Poincaré duals of cohomology generators.
result An explicit description of the Poincaré dual of each generator of the rational cohomology ring.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.
problem Analyzing minimal submanifolds in Poincare-Einstein manifolds.
method Deriving formulae for first and second variations of renormalized volume, proving asymptotic descriptions, and deriving inner-product relationships.
result Existence of asymptotic description and L2-inner-product relationship for specific cases. In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the cla…
Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.
problem Understanding 0-instantons on hyperbolic manifolds and their properties.
method Analyzing asymptotic expansions and using Fefferman-Graham expansion for Poincaré-Einstein metrics.
result The 0-instanton obstruction tensor is a conformal invariant related to Weyl curvature, vanishing for smooth 0-instantons.
In this paper we study the affine geometric structure of the graph of a polynomial f∈R[x,y]. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of…
We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to th…
Study the spectrum of Poincaré operator in triaxial ellipsoids.
problem Spectrum of the Poincaré operator in triaxial ellipsoids.
method Microlocal analysis of partial differential equations and polynomial vector fields.
result Polynomial eigenvectors and large-degree asymptotics of the operator.
Let X be a compact Kähler manifold and S a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Kähler metrics on non-compact Kähler manifold X−S with Poincaré--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and …
Let (ρ_\la)_{\la\in \La} be a holomorphic family of representations of a surface group π_1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space \La, t…
Study geodesics of meromorphic connections on Riemann surfaces.
problem Understanding the asymptotic behaviors of geodesics in meromorphic connections.
method Use branched affine structure induced by Fuchsian meromorphic connections.
result Examples of geodesics with infinitely many self-intersections and peculiar omega-limit sets.
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
Analytic torsion defined for surfaces with cusps, linking to non-cusped surfaces.
problem Defining and analyzing analytic torsion for surfaces with cusps.
method Defining analytic torsion for surfaces with cusps and proving anomaly formula.
result Analytic torsion for surfaces with cusps related to non-cusped surfaces.
Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle 6π. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …
We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincaré-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.
We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist…
CMS formulation solves Poincare conjecture for all dimensions.
problem Poincaré Conjecture in higher dimensions.
method Calculus of moving surfaces (CMS) for evolving hypersurfaces.
result Compact simply connected hypersurfaces relax to constant mean curvature (CMC) manifolds.
Study Bergman kernel and Kähler metrics on Riemann surfaces and symmetric products.
problem Estimating metrics on Riemann surfaces and symmetric products.
method Investigates Bergman kernels and Kähler metrics on Riemann surfaces and symmetric products.
result Estimates the Bergman metric and Kähler metric on symmetric products in terms of the Bergman kernel and Poincaré metric.
Solves an Arnold trivium problem using calculus and topology.
problem Finding critical points on a two-dimensional surface.
method Lagrange multipliers, Morse theory, Poincare-Hopf theorem.
result Determines the genus of a two-dimensional surface.
Study on counting orbits and Poincaré series for specific hyperbolic metrics.
problem Counting orbits and analyzing Poincaré series for strongly hyperbolic metrics.
method Combining ergodic theory techniques with topological flows and symbolic dynamics.
result Obtained orbital counting results and described the domain of analyticity for Poincaré series.
Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.
problem Estimating eigenvalues of the p-Laplacian on submanifolds of asymptotically hyperbolic manifolds.
method Sharp upper and lower bounds derived using conformal techniques and properties of submanifolds.
result Lower bounds on the first eigenvalue for minimal and bounded mean curvature submanifolds.
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu−e2u=K0(z) on general open surfaces. A few oth…
Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.
problem Linking number of Legendrian knots on negatively curved surfaces.
method Analyzes Poincaré series on negatively curved surfaces.
result Explicit rational value of Poincaré series at 0 interprets linking number of Legendrian knots.
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
The paper measures non-convexity of real algebraic curves near a strict local minimum.
problem Measuring the non-convexity of real algebraic curves near a strict local minimum.
method Introduced a new combinatorial object, the Poincare-Reeb graph, to encode and quantify the shape of curves.
result The Poincare-Reeb graph is a plane tree and can be used to study the asymptotic behaviour of level curves near a strict local minimum.
Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
problem Rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.
method Established ε-regularity for Weyl curvature and proved rigidity results.
result Any Poincaré-Einstein filling of S1imesSn−1 must be hyperbolic if non-positively curved. This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This giv…
Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bund…