We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…
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Singular minimal translation surfaces in Euclidean spaces endowed with semi-symmetric connectionsmath.DG
Study of minimal surfaces in 3D space with special connections.
problem Classification of minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
method Analysis of singular minimal translation surfaces in a 3D Euclidean space with a semi-symmetric connection.
result Classification of singular minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
Study projective connections on surfaces using osculating spaces.
problem Understanding projective connections on surfaces.
method Analyzing the second neighborhood and osculating behavior of integral curves.
result Geometrical interpretation of joint invariants of a related group.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy growth rate.
In math.SG/0303255, we discussed the connected components of the space of surface group representations for any compact connected semisimple Lie group and any closed compact (orientable or nonorientable) surface. In this sequel, we generalize the results in math.SG/0303255 in two directions: we consider general compact…
The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
On d-Holomorphic Connectionsmath.AG
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
The study classifies minimal translation surfaces in 3D and 3D_1.
problem Classifying minimal translation surfaces in specific geometric settings.
method Defined and classified minimal translation surfaces with semi-symmetric connections.
result New classification of minimal translation surfaces in and .
Criterion for Lie algebroid connections on compact Riemann surfaces.
problem Finding conditions for Lie algebroid connections on compact Riemann surfaces.
method Analyzing stable holomorphic vector bundles and their connections.
result Necessary and sufficient condition for Lie algebroid connections on compact Riemann surfaces.
New minimal surfaces found using a modified metric connection.
problem Finding minimal surfaces in Euclidean 3-space.
method Used a special semi-symmetric metric connection instead of the Levi-Civita connection.
result Found non-trivial minimal surfaces other than planes.
Embedded surfaces in a ball have any genus and connected boundary.
problem Existence of embedded free boundary minimal surfaces with specific properties.
method Min-max techniques applied to the unit ball in .
result Existence of embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
This paper classifies fundamental groups of perforated surfaces.
problem Classifying the fundamental groups of perforated surfaces.
method Using the classification theorem for surfaces and covering spaces.
result Fundamental groups of perforated surfaces are large and not Hopfian.
Study precise rates of horizontal gap shrinkage on generic translation surfaces.
problem Understanding precise decay rates of horizontal gaps in translation surfaces.
method Analyzing saddle connections and their angles on translation surfaces.
result Obtained precise decay rates for the difference in angle between almost horizontal saddle connections.
Study extends Yang-Mills energy gap to Kähler surfaces.
problem Extend Yang-Mills energy gap to Kähler surfaces.
method Extend -energy gap to compact Kähler surfaces with generic Kähler metrics.
result All ASD connections on principal bundles over Kähler surfaces are irreducible.
Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.
problem Existence and properties of surfaces with specific canonical and geometric genus conditions.
method Study of rational curve configurations and use of -Gorenstein smoothings.
result Existence of -dimensional families of simply-connected surfaces with and .
Study on surfaces with constant curvature under a specific connection.
problem Classifying surfaces with constant sectional curvature under a semi-symmetric non-metric connection.
method Analyzing surfaces in with a canonical semi-symmetric non-metric connection determined by a vector field.
result Classification of surfaces under various conditions (cylindrical, rotational) with constant sectional curvature.
Optimal connectivity results for spheres in the curve graph of low and medium complexity surfacesmath.GT
The study connects spheres in specific surface curve graphs, proving connectivity and classifying components.
problem Proving connectivity and classifying components of spheres in curve graphs of low and medium complexity surfaces.
method Analyzing specific surfaces and , proving connectivity and classifying components.
result Spheres of any radius are connected in , and the union of two consecutive spheres is connected in and .
The paper classifies translation surfaces with constant curvature in a specific connection.
problem Classifying translation surfaces with constant curvature in a semi-symmetric non-metric connection.
method Completely classified translation surfaces of constant sectional curvature in a semi-symmetric non-metric connection.
result Translation surfaces of constant curvature are generalized cylinders, similar to the Levi-Civita connection but with additional non-constant curvature cases.
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
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.
Affine connections linked to Riccati distributions on compact surfaces.
problem Understanding affine structures on complex compact surfaces.
method Established a correspondence between affine connections and Riccati distributions.
result One-to-one correspondence between affine structures and Riccati foliations on compact surfaces.
Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong's canonical coordinate form of such connections is simplified.
The paper proves mapping class groups of closed surfaces are simply connected at infinity.
problem Understanding connectivity at infinity for mapping class groups of surfaces.
method Proved a general simply connected at infinity result for finitely presented groups.
result All mapping class groups of closed surfaces of genus ≥ 3 are simply connected at infinity.
Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theorycs.GR
Two novel algorithms for conformal parameterization of multiply-connected surfaces.
problem Parameterization of surfaces with holes.
method Developed efficient methods using quasi-conformal theory.
result Efficient conformal parameterization of multiply-connected surfaces.
Classifies meromorphic affine connections on complex surfaces.
problem Investigating uniformization in higher dimensions with singularities.
method Extending work on holomorphic connections, classifying meromorphic connections on compact surfaces.
result Classification of meromorphic affine connections on compact complex surfaces.
SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…
Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
problem Characterizing holomorphic Lie algebroid connections on Riemann surfaces.
method Analyzes conditions for holomorphic vector bundles to admit Lie algebroid connections based on Lie algebroid properties.
result Conditions for holomorphic vector bundles to admit holomorphic Lie algebroid connections are determined.
Constructs irreducible flat connections on a Riemann surface.
problem Creating flat connections with specific monodromy on Riemann surfaces.
method Constructs irreducible holomorphic connections with SL(2,R) monodromy.
result Answers a question about flat connections on Riemann surfaces.
The study shows how to measure translation surfaces with short saddle connections.
problem Measuring the probability of surfaces with short saddle connections.
method Using the multi-scale compactification of strata and algebraicity results.
result Proves strong regularity for invariant measures on translation surfaces.
Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.
problem Characterize dilation surfaces with dense hyperbolic geodesics.
method Analyzing saddle connections and directional flow properties.
result Dilation surfaces with horizon saddle connections have dense hyperbolic geodesics.
To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …
We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every pr…
Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain non-homotopic loops.
We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic…
Study flip graphs for surfaces of infinite type, finding uncountably many connected components.
problem Understanding relationships between triangulations of infinite type surfaces via flips.
method Associate triangulations to flip graphs and study sequences of simultaneous flips.
result Flip graphs for infinite type surfaces have uncountably many connected components.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let be …
Flat surfaces that correspond to -differentials on compact Riemann surfaces are of finite area provided there is no pole of order or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a -differential with at least one pole of order at least $k…
Kählerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvaturesmath.DG
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relatio…
Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the -action is ergodic in co…
This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves, nonseparating curves, pants, and cut systems are all connected for genus $g \gg…
Improved method for numerical conformal mappings on complex domains.
problem Accurate and efficient computation of conformal mappings on multiply connected domains.
method Generalization and refinement of the conjugate function method using high-order finite element methods.
result Achieved accurate and efficient construction of boundary values for multiply connected domains.
We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the …
New methods connect low-loss points on neural network surfaces.
problem Connecting low-loss points on neural network loss surfaces.
method Macroscopic distributional assumptions and global connection models.
result Accuracy correlates with complexity and sensitivity.