After a short summary of known results on surface-complexity of closed 3-manifolds, we will classify all closed orientable 3-manifolds with surface-complexity one.
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35 results for “surface-complexity”
The paper introduces surface-complexity to measure 3-manifold complexity.
problem Measuring the complexity of compact 3-manifolds.
method Defining surface-complexity via Dehn surfaces and proving its properties.
result Surface-complexity equals the minimal number of cubes in a cubulation for certain manifolds.
We define the surface complex for -manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex is…
Classifies 3-manifolds from cube identifications.
problem Classifying non-orientable 3-manifolds.
method Identifying cube faces to form manifolds, classifying based on surface-complexity.
result Identifies four flat non-orientable 3-manifolds with surface-complexity one.
We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on …
Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.
problem Classifying semi-algebraic surfaces with isolated singularities.
method Bi-Lipschitz homeomorphisms with inner distance.
result Complete classifications for Nash surfaces and complex algebraic curves.
4-manifolds can be broken down into pairs-of-pants and K3 surfaces.
problem Decomposing 4-manifolds diffeomorphic to complex hypersurfaces.
method Pair-of-pants and K3 surfaces decomposition.
result 4-manifolds diffeomorphic to complex hypersurfaces can be decomposed into a specific number of pair-of-pants and K3 surfaces.
Study minimal surfaces in complex hyperbolic space, linking entropy and volume.
problem Characterize minimal submanifolds in complex hyperbolic space.
method Analyze asymptotic regularity and introduce Colding-Minicozzi entropy and CR-volume.
result Establish a connection between Colding-Minicozzi entropy and CR-volume.
The paper modifies twistor spaces for Kähler surfaces and finds metrics on the modified spaces.
problem Constructing a modified twistor space for Kähler surfaces and studying its properties.
method Constructing a modification of the twistor space of a Kähler scalar flat surface and studying its complex-geometric and metric properties.
result Complete balanced metrics are constructed on and it is shown that cannot be Kähler when is a compact simple hyperkähler manifold.
We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if is a generic connection on a princ…
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space formsmath.DG
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
Flexible Surfaces in and math.GT
Flexible surfaces found in complex projective and product spaces.
problem Finding flexible surfaces in complex projective and product spaces.
method Constructing flexible surfaces within prescribed homology classes.
result Flexible surfaces exist in both and .
New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
The note proves a metric equivalence for stable bundles on surfaces.
problem Understanding stability conditions and metrics on complex projective surfaces.
method Analyzing stability in the large scaling limit and proving equivalence with deformed Hermitian-Yang-Mills metrics.
result Equivalence of stability and deformed Hermitian-Yang-Mills metrics for smooth projective surfaces.
Complex analysis aids in studying minimal surfaces.
problem Understanding minimal surfaces in Euclidean spaces.
method Complex-analytic techniques applied to conformal minimal surfaces.
result New results on approximation, interpolation, and general position properties.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
The study finds minimal surfaces in complex space forms are often totally geodesic.
problem Characterizing minimal surfaces with specific geometric properties in complex space forms.
method Analyzing free-boundary minimal surfaces in geodesic balls of complex space forms.
result Minimal surfaces in certain complex space forms are either totally geodesic or superminimal.
Algorithm classifies surface homeomorphisms with polynomial time complexity.
problem Classifying surface homeomorphisms with polynomial time complexity.
method Algorithm to compute curve distances and decide Nielsen-Thurston types.
result Polynomial time classification of surface homeomorphisms.
Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.
problem Proving the existence of CMC-1 surfaces with arbitrary complex structures in hyperbolic space.
method Using a jet interpolation theorem and a uniform approximation theorem for holomorphic null curves.
result Existence of complete densely immersed CMC-1 surfaces in hyperbolic space with arbitrary complex structure.
Exotic surfacesmath.AG
The paper constructs exotic complex projective surfaces using rational blowdowns.
problem Finding exotic complex projective surfaces with specific invariants.
method Rational blowdowns and construction of Kollár--Shepherd-Barron--Alexeev surfaces.
result First exotic surfaces constructed, including minimal and symplectic ones.
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Study on positivity properties of vector bundle Monge-Ampère equation.
problem Analyzing positivity in vector bundle Monge-Ampère equation.
method Investigates MA-positivity and MA-semi-positive solutions for different ranks of holomorphic bundles over complex surfaces and manifolds.
result Positivity preservation in rank-two holomorphic bundles but not in higher ranks.
Clustering is a fundamental problem in many scientific applications. Standard methods such as -means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of -means and hierarchical clustering s…
The paper extends a theorem about stable minimal surfaces to higher codimensions.
problem Stability and holomorphicity of parabolic stable minimal surfaces in higher-dimensional spaces.
method Generalization of a classical theorem to higher codimensions, with additional assumptions on the normal bundle.
result Holomorphicity of stable minimal surfaces in higher-dimensional spaces.
Researchers determine all possible representations of monodromy for Schwarzian equations on punctured surfaces.
problem Determine representations of monodromy for Schwarzian equations on punctured surfaces.
method Explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy.
result All possible representations of monodromy for Schwarzian equations on punctured surfaces are determined.
Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.
problem Understanding the space of complex projective structures on surfaces with circle patterns.
method Analyzing ideal polyhedral surfaces in hyperbolic ends, proving manifold properties and Lagrangian immersions.
result The space of complex projective structures on surfaces with circle patterns is a manifold of dimension 6g-6.
New methods show surfaces in HNN extensions have complexity at least their boundary complexity.
problem Complexity of surfaces in HNN extensions and nontriviality of one-relator quotients.
method Stable commutator length and HNN extensions.
result Surfaces in certain HNN extensions have complexity no less than their boundary complexity.
Study para-hyperKähler geometry of anti-de Sitter structures.
problem Understand the geometry of anti-de Sitter structures.
method Investigate para-hyperKähler structures and their relations.
result Found neutral pseudo-Riemannian metric and symplectic structures.
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 is monotonically increasing in genus .
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in under certain conditions.
We simplify Thurston norm computation for 2-bridge link complements.
problem Understanding the complexity of Thurston norm unit balls in 3-manifolds.
method Utilized Floyd and Hatcher's surface description and integral class minimization.
result Thurston norm unit balls of 2-bridge link complements have at most 8 faces.