In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…
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48 results for “degenerate surfaces”
The paper discovers new ways Riemann surfaces can degenerate.
problem Understanding degeneration of infinite-type Riemann surfaces.
method Constructing a concrete example to prove degeneration phenomena.
result Existence of degenerations in Bers boundary of infinite-type surfaces.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
Characterizes Wahl singularities in del Pezzo surface degenerations.
problem Classifying Wahl singularities in degenerations of del Pezzo surfaces.
method Introducing del Pezzo Wahl chains with markings, proving degenerations to toric surfaces, establishing correspondences, and using Hacking's exceptional collections.
result Established a one-to-one correspondence between marked del Pezzo surfaces and fake weighted projective planes.
Defines height pairing for differential forms on Riemann surface degenerations.
problem Calculating heights for differential forms on degenerating Riemann surfaces.
method Defines Archimedean height pairing, uses Dai-Yoshikawa asymptotics, extends Filip-Tosatti construction.
result Relates new pairing to current-valued pairing, extends geometric settings.
In this paper we study the degeneration of convex real projective structures on bordered surfaces.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfacesmath.DG
Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
Researchers prove smoothings for surfaces with triple points.
problem Smoothings of surfaces with triple points.
method Differential geometric proof.
result Proves existence of smoothings for surfaces satisfying suitable conditions.
Study higher genus polylogarithms under Riemann surface degenerations.
problem Understanding higher genus polylogarithms under degenerations.
method Investigate the Enriquez connection for polylogarithms and show it becomes a known connection for families of Riemann surfaces.
result Higher genus polylogarithms can be described explicitly as power series in deformation parameters and logarithms of families.
The paper classifies degenerate almost complex surfaces in a nearly Kähler space.
problem Classifying degenerate almost complex surfaces in nearly Kähler spaces.
method Investigates two distinct cases based on the preservation of the tangent bundle under the almost product structure.
result Complete and explicit classification of degenerate almost complex surfaces in nearly Kähler spaces.
Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.
problem Compare and study surfaces with opposite Gaussian curvatures under two different metrics.
method Consider surfaces in homogeneous 3-manifolds with two metrics, impose extrinsic curvature conditions, and analyze Gaussian curvature functions.
result Identify and characterize anisocurved surfaces with opposite Gaussian curvatures under both metrics.
Lectures on symplectic aspects of surface degenerations at KIAS.
problem Exploring symplectic structures in surface degenerations.
method Expository account of symplectic aspects of cyclic quotient surface singularities.
result Discussion of symplectic structures in surface degenerations.
We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars a…
Classifies normal stable Horikawa surfaces with smoothable singularities.
problem Characterizing surfaces with specific singularities and smoothability criteria.
method Classification and smoothability criterion based on log canonical singularities.
result Provides a criterion for global -Gorenstein smoothability of Horikawa surfaces.
We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.
New PL invariant classifies K3 surface degenerations.
problem Classifying type II degenerations of K3 surfaces.
method Explicit PL convex function from interval, differential geometric viewpoint.
result Function classifies degenerations into combinatorial types.
Defines log Floer cohomology for symplectic surfaces with a degenerate part.
problem Extending Floer cohomology to degenerate symplectic structures.
method Definition of log Floer cohomology for oriented log symplectic surfaces.
result Log Floer cohomology is invariant under isotopies and isomorphic to log de Rham cohomology for a single Lagrangian.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
Study degenerations of Kähler-Einstein metrics on surfaces.
problem Understanding the geometry of Kähler-Einstein metrics on surfaces as they degenerate.
method Construct a Kähler-Einstein neck region to model degeneration.
result Provides a model for the limiting geometry of metrics in the family.
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifoldsmath.DG
A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…
The paper studies hanging chains and surfaces in degenerate geometries.
problem Investigating hanging chains and surfaces in simply isotropic plane and space.
method Characterizing catenaries and proving them as minimal surfaces in the simply isotropic space.
result The simply isotropic catenary is the generating curve of a minimal surface of revolution.
Study small eigenvalues of Riemann surfaces degenerating with Kähler metrics.
problem Determining small eigenvalues of the Laplacian on degenerating Riemann surfaces.
method Combining heat kernel estimates and Quillen metrics to compute asymptotic behavior of eigenvalues.
result Explicit calculation of small eigenvalues as a function of the parameter.
We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in and modulo bubbles of sequences of such maps.
Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
It is constructed a normal form for a class of real-smooth surfaces M\subset\mathbb{C}^{2} defined near a degenerate CR singularity.
The paper studies how hyperbolic surfaces degenerate along harmonic map rays.
problem The degeneration of hyperbolic surfaces along harmonic map rays.
method Using Teichmüller space and holomorphic quadratic differentials, the authors show convergence of rescaled distance functions to the intersection number with a vertical measured foliation.
result Hyperbolic surfaces along the ray converge to the dual R-tree of the vertical measured foliation in the sense of Gromov-Hausdorff.
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.
A connected regular surface in Lorentz-Minkowski 3-space is called a mixed type surface if the spacelike, timelike and lightlike point sets are all non-empty. Lightlike points on mixed type surfaces may be regarded as singular points of the induced metrics. In this paper, we introduce the L-Gauss map around non-degener…
We consider the first non-zero eigenvalue of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve previous …
Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine…
We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu-Schwarz type nodes. We find condition that is both necessary and sufficient for the …
The paper studies Möbius inversion on surfaces in Minkowski 3-space.
problem Understanding transformations of surfaces in Minkowski space.
method Definition and properties of Möbius inversion on surfaces in Minkowski 3-space.
result Möbius inversion preserves lines of principal curvature and degenerate metric points but not parabolic sets.
Study on spectral gaps of hyperbolic surfaces as genus increases.
problem Understanding differences in eigenvalues for large genus hyperbolic surfaces.
method Analysis of the Laplacian on degenerating hyperbolic surfaces, min-max principle.
result Supremum of spectral gaps has infimum limit of at least 1/4 as genus increases.
Study continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all n…
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
Study on Dirac operator spectrum on shrinking surfaces with cusps.
problem Behavior of Dirac operator spectrum on degenerating Riemannian surfaces.
method Adapted pseudodifferential calculus, including Dirac operators and their resolvents.
result Smoothness of spectral projectors and regularity for the cusp-surgery trace.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
Study of Steklov eigenvalues on degenerating conformal classes.
problem Understanding Steklov eigenvalues on surfaces with boundaries.
method Precise formula for the limit of Steklov eigenvalues on degenerating conformal classes.
result The limit of Steklov eigenvalues equals for surfaces with boundaries.
A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field . Several sufficient assumptions on such a surface with non-degenerate -second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…
In this paper we study the deformation problem of pairs consisting of a Riemann surface and a holomorphic line bundle over that surface, and also sections thereof. We emphasize a constructive approach throughout and work and use covering space techniques. In particular, we also describe the limits of such degenerations…
New insights into surface energy reduction.
problem Energy behavior of degenerating submanifolds.
method Analyzing regularized Riesz energy for closed submanifolds.
result Energy blows up as submanifolds degenerate.
Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
We present new rectification theorems of degenerate quasi-conformal structures that give a meaning to quotients of Riemann surfaces with empty interior "fundamental domains". These techniques are used to define the unique renormalization of polynomials with Cantor set Julia sets.