Researchers exhaust curve graph using rigid expansions on surfaces.
problem Exhausting the curve graph of surfaces with genus ≥ 3.
method Constructing a finite set of curves and using iterated rigid expansions.
result The constructed set exhausts the curve graph via rigid expansions.
Paper develops a new algorithm to find shortest paths on surfaces.
problem Finding shortest paths on surfaces with defined metrics.
method Uses Taylor expansion of exponential map for numerical computation.
result Developed a new algorithm to find geodesics efficiently.
Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.
problem Understanding geodesic flows on compact surfaces without conjugate points.
method Time-preserving semi-conjugation to a continuous expansive flow.
result Geodesic flows on compact surfaces without conjugate points of genus > 1 have a unique measure of maximal entropy.
Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.
problem Blowup behavior of regularized solutions to Jang equation inside apparent horizons.
method Two geometric treatments: dilation and translation. Characterization of limits of rescaled and translated solutions.
result Limits of properly rescaled solutions are constant expansion surfaces.
In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in R^n up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R^3. Also we compute, by using the Taylor expan…
Study conical Ricci flow on surfaces with explicit asymptotic expansions.
problem Analyzing Ricci flow on conical surfaces with explicit regularity.
method Established framework for linear parabolic equations on conical surfaces; proved long-time existence and optimal regularity of conical Ricci flow.
result Explicit asymptotic expansions of conformal factor for conical Ricci flow.
Study local foliations of surfaces with constant mean curvature and constant expansion in space-time.
problem Characterize surfaces with constant mean curvature and constant expansion in space-time.
method Use Lyapunov Schmidt reduction in an n+1 dimensional manifold to construct and prove the uniqueness of foliations.
result Construct and prove the uniqueness of local foliations of surfaces with constant mean curvature and constant expansion.
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.
problem Understanding random representations of surface groups into special unitary groups.
method Use of a symplectic form on moduli space, establishing asymptotic expansions for trace values.
result Existence of large n asymptotic expansions for expected values of trace of elements under random representations.
New method detects geometric intersection number greater than zero for curves on surfaces.
problem Detecting geometric intersection number greater than zero for curves on surfaces.
method Computing a value in the first homology group using elements of the fundamental group and Dehn twist.
result Explicit formula for Dehn twist action on free groups provides effective tool.
Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.
problem Analyzing the volume of moduli spaces of hyperbolic surfaces with varying genus.
method Topological recursion formula by Mirzakhani, asymptotic expansion for high genus.
result Explicit computation of the second term in the asymptotic expansion.
The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the Baker-Campbell-Hausdorff series.
This paper exhausts curve complexes on non-orientable surfaces.
problem Proving exhaustion of curve complexes on non-orientable surfaces.
method Proving exhaustion via rigid expansions and graph endomorphisms.
result Any graph endomorphism of curve complexes whose restriction to a finite rigid set is injective is induced by a homeomorphism.
Study solutions to metric equations on modified surfaces.
problem Finding constant scalar curvature Kähler metrics on modified surfaces.
method Expanding solutions to extremal metric type equations.
result Developed methods to solve metric equations on modified surfaces.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
This study exhausts curve graphs of low-genus surfaces.
problem Exhausting curve graphs of low-genus surfaces.
method Constructing finite subgraphs and using rigid expansions.
result Graph morphisms and endomorphisms are automorphisms and induced by homeomorphisms.
Study the spectral geometry of surfaces with curved conic singularities.
problem Understanding the spectral properties of surfaces with conic singularities.
method Using the heat trace expansion, express spectral geometry terms through the geometry and curvature of the singularities.
result The first few terms in the heat trace expansion are expressed through the geometry and curvature of the singularities.
We establish multiparameter resolvent trace expansions for elliptic boundary value problems, polyhomogeneous both in the resolvent and the auxiliary parameter. The present analysis is rooted in the joint project with Matthias Lesch on multiparameter resolvent trace expansions on revolution surfaces with applications to…
Study of Witten-Reshetikhin-Turaev invariants for mapping tori.
problem Asymptotic expansion of Witten-Reshetikhin-Turaev invariants of mapping tori.
method Geometric quantization of moduli spaces of flat connections, Picard-Lefschetz theory for Laplace integrals.
result Full asymptotic expansion provided for pseudo-Anosov mapping classes on a punctured torus.
The study generalizes Bochner Laplacian results to Riemann surfaces.
problem Analyzing curvature vanishing line bundles on Riemann surfaces.
method Exploiting the relation of Bochner Laplacian on tensor powers with sR Laplacian.
result Bergman kernel expansion for semi-positive line bundles.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
Paper proves existence of minimal surfaces with alternating multiple zeta values.
problem Existence and properties of minimal surfaces.
method Complex analytic methods to deform Lawson surfaces.
result Area of minimal surfaces ξ1,g is monotonically increasing in genus g. For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increas…
Geometric quantization results for Riemann surfaces with semi-positive line bundles.
problem Analyzing geometric quantization for Riemann surfaces with semi-positive line bundles.
method Exploring the Bergman kernel expansion and related results for induced Fubini-Study metrics, Toeplitz operators, and holomorphic torsion.
result Asymptotic results for holomorphic torsion and random sections.
Sharp eigenvalue estimates on degenerating hyperbolic surfaces.
problem Estimating the first non-zero eigenvalue of Laplacian on hyperbolic surfaces as a collar degenerates.
method Using the relationship between the eigenvalue and Fenchel-Nielsen length coordinate, proving estimates with optimal error rates.
result Improved estimates and new information on leading order terms of eigenvalue expansion.
Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.
problem Computing the exact asymptotic value of Weil-Petersson volumes for large genus surfaces.
method Analysis of Witten-Kontsevitch intersection numbers and expansion of volumes.
result Exact asymptotic value of volume polynomials computed for hyperbolic surfaces.
The paper studies magnetic field effects on surface eigenvalues and spectral properties.
problem Understanding magnetic effects on surface eigenvalues and spectral properties.
method Provided precise spectral asymptotics expansion for the magnetic Dirichlet-to-Neumann map on surfaces.
result The spectrum of the magnetic Dirichlet-to-Neumann map uniquely determines the number and length of boundary components, parallel transport, and magnetic flux.
For any strictly positive martingale S=exp(X) for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing t…
Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.
problem Asymptotic expansion of graph Laplacian on discretized surfaces.
method Relate spanning trees and cycle-rooted spanning forests to zeta-regularized determinants.
result Explicit formula for limit of cycle-rooted spanning forest probability and topological observables.
Extends results on marginally outer trapped surfaces to general null expansion.
problem Analyzing geometry and topology of expanding horizons.
method Introduces g-stability and proves conditions for positive Yamabe type and scalar curvature. result Initial data sets with compact boundary of positive null expansion have positive mass.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…
Maps preserve edges between curve graphs of surfaces with certain properties.
problem Proving homeomorphism between surfaces based on curve graphs.
method Using simplicial properties of rigid expansions.
result Edge-preserving maps imply homeomorphic surfaces.
Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.
problem Understanding the behavior of harmonic maps from surfaces to homogeneous spaces, especially in the presence of bubbles.
method Refined asymptotic expansions and obstruction relations for sequences developing a single bubble, geometric constraints for weakly conformal maps.
result New geometric constraints on the tangent planes of the limit map and bubble, depending on the dimensionality.
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.
New volume functions for random hyperbolic surfaces link to spectral gaps.
problem Analyzing spectral gaps in random hyperbolic surfaces.
method Introduced new volume functions VgT(l), derived their asymptotic expansions, and linked them to spectral gaps. result Coefficients in the asymptotic expansion of VgT(l) are Friedman-Ramanujan functions. The paper studies finite TYCZ expansions on Kaehler manifolds and their relation to cscK metrics.
problem Finite TYCZ expansions on Kaehler manifolds and their connection to cscK metrics.
method Analyzes finite TYCZ expansions on Kaehler manifolds and their properties.
result Finite TYCZ expansions imply polynomial behavior of certain metrics and vanishing of log-term in Szegö kernel.
The paper expands cluster algebra formulae to non-orientable surfaces and proves positivity.
problem Proving positivity for quasi-cluster algebras from non-orientable surfaces.
method Generalizing Musiker, Schiffler, and Williams' expansion formulae to principal laminations and quasi-triangulations.
result Positivity for quasi-cluster algebras is proven with respect to any choice of coefficients.
Let M be a regular Riemann surface with a metric which has constant scalar curvature ρ. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles KMm. That is, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\fracρ{2 m})+O(e^{-\frac{(\log m)^{2}…
We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing struc…
We prove the expansion formula for the classical Futaki invariants on the blowup of Kähler surfaces, which explains the balancing condition of Arezzo-Pacard. The relation with Stoppa's result is also discussed.
Deformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation ex…
Study of umbilic points on Willmore surfaces in 3-sphere.
problem Characterizing umbilic points on Willmore surfaces.
method Analysis of conformal Gauss map and Gauss-Bonnet formula.
result Unified expression for Willmore energy in space-forms.
The study examines metrics on Riemann moduli spaces and their asymptotic expansions.
problem Analyzing metrics on Riemann moduli spaces and their behavior near exceptional divisors.
method Finding complete asymptotic expansions of Weil-Petersson and fiber metrics using hyperbolic metrics on fibers and push-forward theorem for conormal densities.
result Complete asymptotic expansions of metrics on Riemann moduli spaces near exceptional divisors.
We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.
Study on quantum Hall effect using Riemann surfaces and Quillen metric.
problem Quantum Hall effect and its adiabatic transport coefficients.
method Analyzing generating functional, adiabatic curvature, and Quillen metric.
result Identified Quillen metric as non-local part of generating functional expansion.
Study counts geodesics on modular surface, linking to necklace counting.
problem Counting geodesics on modular surface with specific winding numbers.
method Asymptotic expansion, generating function analysis, correspondence to necklace counting.
result Obtained asymptotic growth rate of m low-lying geodesics in terms of word length.