Topological field theory computes Arf invariant of r-spin surfaces.
problem Computing the Arf invariant of r-spin surfaces.
method Combinatorial model of r-spin surfaces, state sum construction of topological field theories.
result The r-spin Arf invariant is constant on orbits of the mapping class group.
We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…
We give a review of the quantum singularity theory of Fan-Jarvis-Ruan and the r-spin theory of Jarvis-Kimura-Vaintrob and describe the work of Abramovich-Jarvis showing that for the singularity A_{r-1} = x^r the stack of A_{r-1}-curves of is canonically isomorphic to the stack of r-spin curves. We prove that the A_{r-1…
We introduce the stack of r-spin maps. These are stable maps into a variety V from n-pointed algebraic curves of genus g, with the additional data of an r-spin structure on the curve. We prove that this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We s…
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using r-spins and investigation of self-map degrees. result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
Classifies components of abelian differentials over Teichmüller space.
problem Classifying components of strata of abelian differentials.
method Computing monodromy groups and determining finite generating sets for r-spin stabilizer subgroups. result Complete classification of strata components for g≥5. Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σare r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle STΣ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomo…
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of r-spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a …
String-net models explore non-spherical fusion categories, revealing new spin structures and representations.
problem Investigating string-net models in non-spherical fusion categories.
method String-net models associate vector spaces to surfaces in terms of graphs decorated by objects and morphisms of a pivotal fusion category.
result String-net spaces count r-spin structures and carry representations of the mapping class group.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r-spin and Theta-class intersection numbers. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_…
Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut…
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
problem Formulating and proving a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for tetrahedra.
method Formulated and proved a Lorentzian analogue of Minkowski theorem for tetrahedra in dS3 and AdS3.
result A unique strictly convex tetrahedron can be reconstructed from four non-trivial based SO+(1,2) holonomies.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
The study characterizes polynomial conserved quantities for Lie applicable surfaces.
problem Characterizing polynomial conserved quantities for Lie applicable surfaces.
method Gauge theoretic approach for Lie applicable surfaces, including isothermic, Guichard, and L-isothermic surfaces. result Induced transformations of Lie applicable surfaces for well-known transformations and new Bäcklund-type transformation for linear Weingarten surfaces.
Study on focal surfaces of tubular surfaces in 3D space, focusing on their flatness and asymptotic properties.
problem Characterizing and understanding focal surfaces of tubular surfaces in 3D space.
method Defined tubular surfaces using Frenet and Darboux frames, analyzed their focal surfaces, and derived conditions for flatness.
result No minimal focal surface exists in 3D space for tubular surfaces.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
Stable-ribbon surface-links have unique handle-irreducible summands.
problem Characterize handle-irreducible summands of stable-ribbon surface-links.
method Combining old results on stably trivial surface-links and surface-knots with infinite cyclic fundamental groups.
result Every handle-irreducible summand of a stably trivial surface-link is a trivial 2-link.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. New surfaces in 4-ball constructed from knits, described by charts.
problem Constructing surfaces in 4-ball from knits.
method Introducing knitted surfaces, describing them with BMW charts.
result Every compact surface in 4-ball is ambiently isotopic to a knitted surface.
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
New surfaces generalize Dini surfaces in 4D.
problem None explicitly stated; focuses on surface generalization.
method Introducing a new family of surfaces in 4D.
result Generalized Dini surfaces exist in 4D.
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Minimal surfaces are the only biharmonic in Sol3.
problem Characterizing biharmonic surfaces in Sol3.
method Found local equations for biconservative surfaces and showed all biharmonic surfaces are minimal.
result All biharmonic surfaces in Sol3 are minimal.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
New surface class defined using osculating circles.
problem Defining a new surface class in Euclidean space.
method Using osculating circles of curves and classification of specific types.
result Classification of canal and Weingarten surfaces.
Classifies surfaces in specific 3-manifolds with isometry group.
problem Classifying surfaces in homogeneous 3-manifolds.
method Explicit classification of four families of surfaces.
result Classification of surfaces in homogeneous 3-manifolds.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
The Enneper surface and helix surfaces are unique in their geometric properties.
problem Characterizing surfaces with specific geometric properties.
method Analyzing isogonal lines and pseudo-geodesic lines in 3D Euclidean space.
result Helix surfaces and Enneper surface are the only surfaces with isogonal lines as generalized helices and pseudo-geodesic lines.
We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…
Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.
problem Understanding the properties of least area surfaces in 3-manifolds.
method Introducing quasi-normal surfaces and showing their relationship to least area surfaces in fine triangulations.
result Least area surfaces in 3-manifolds are quasi-normal with respect to fine triangulations, and this quasi-normality leads to piecewise flat approximations.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.