Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
Method resolves 4D symplectic orbifolds using complex geometry.
problem Resolving symplectic orbifolds in 4 dimensions.
method Combining complex geometry techniques with symplectic form gluing.
result Examples of 4D symplectic orbifolds successfully resolved.
Study on 4D PDEs with half-flat conformal structure leading to Monge-Ampere equations.
problem Characterizing second-order PDEs in 4D with specific conformal structures.
method Analysis of Monge-Ampere property and dispersionless Lax pairs.
result All known scalar second-order integrable dispersionless PDEs in 4D are of Monge-Ampere type.
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.
problem Classifying fibrations of compact flat orbifolds.
method Developed theory for classifying fibrations up to affine equivalence.
result Classified fibrations of compact flat 2-orbifolds.
Study of flat metrics on orbifolds and their moduli spaces.
problem Understanding flat metrics on orbifolds and their moduli spaces.
method Analysis of Teichmüller spaces and mapping class groups.
result Moduli space of flat metrics on orbifolds is a very good orbifold under certain conditions.
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the…
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Proves orbifold singularities for Ricci-flat metrics on certain Kähler varieties.
problem Regularity of singular Ricci-flat Kähler metrics on Kähler varieties with log terminal singularities.
method Analyzes orbifold singularities of metrics restricted to the orbifold locus.
result Singular Ricci-flat Kähler metrics on Kähler varieties with log terminal singularities have orbifold singularities.
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact …
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by col…
In this paper, we establish an equality between the analytic torsion introduced by Dar\cite{MR876230} and the orbifold analytic torsion defined by Ma \cite{MR2140438} on a compact manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an ho…
Study on counting flat connections over G2-orbifolds, proving moduli space compact and smooth.
problem Counting flat connections over G2-orbifolds. method Analyzing moduli space of G2-instantons on flat bundles over torsion-free G2-orbifolds. result Proves moduli space compact and smooth at the irreducible locus.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Flat minimal hypersurfaces in 4D space are always flat.
problem Understanding stable minimal hypersurfaces in 4D space.
method Proving stability and completeness lead to flatness.
result Complete, stable minimal hypersurfaces in 4D are flat.
The paper characterizes Eguchi-Hanson space and its higher-dimensional analogs using Lichnerowicz Laplacian.
problem Characterizing complete Ricci-flat ALE orbifolds.
method Analytical proof using Lichnerowicz Laplacian and dimension constraints.
result Uniqueness of Eguchi-Hanson space and its higher-dimensional analogs among Ricci-flat Kähler ALE orbifolds.
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
We show that all closed flat n-manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)-orbifold.
We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics on TS2, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthe…
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product …
In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n-manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n+1)-orbifold is dense in the space of similarity classes of flat metri…
The paper classifies and proves properties of ALE manifolds and orbifolds.
problem Classifying and understanding the structure of ALE manifolds and orbifolds.
method Proving structure theorems and using properties of conformal blow-up and compactification.
result Several classifications and properties of ALE manifolds and orbifolds are given in low dimensions.
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…
We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti…
Study of orbifold Chern character using superconnections.
problem Chern character for orbifold settings.
method Use of flat antiholomorphic superconnections and Riemann-Roch-Grothendieck theorem.
result Uniqueness of orbifold Chern character proven.
We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal str…
The paper connects orbifold singularities to higher symmetries in SQFTs.
problem Understanding higher symmetries in supersymmetric quantum field theories.
method Cutting and gluing of orbifold singularities to determine symmetries.
result Local orbifold singularities encode 0-form, 1-form, and 2-group symmetries.
New obstructions found for smooth desingularization of compact Einstein orbifolds.
problem Finding obstructions to desingularizing compact Einstein orbifolds.
method Identifying new obstructions specific to compact Einstein 4-orbifolds. result Almost all flat orbifold metrics on T4/Z2 are not limits of Ricci-flat metrics. For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient co…
New invariant for 4D hypersurfaces ensures smooth critical points.
problem Understanding smoothness of curvature energies on 4D hypersurfaces.
method Developed a new conformally invariant energy.
result Critical points of new energy are smooth.
New Ricci flows found with Einstein orbifolds at infinity.
problem Ancient and immortal Ricci flows with Einstein orbifolds at infinity.
method Continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows constructed.
result Found continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows on specific manifolds.
Blowing up flat metrics yields balanced ones with constant curvature.
problem Constructing balanced metrics with constant curvature on orbifolds.
method Blowing up a compact orbifold with balanced Chern-Ricci flat metrics.
result Blown-up orbifolds admit balanced metrics with constant Chern scalar curvature.
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models…
The paper studies special surfaces in 4D space forms with specific geometric properties.
problem Investigating biconservative surfaces with flat normal bundles in 4D space forms.
method Analyzing compatibility conditions, prescribing flat connection, and determining specific surface properties.
result Existence and characterization of biconservative Weingarten surfaces with flat normal bundles.
In this paper we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a 1-orbifold.
The article studies Ricci-flat metrics on complex projective space.
problem Understanding Ricci-flat metrics on complex projective space.
method Description and explicit computation of curvature.
result All compact, minimal submanifolds are contained in the zero section.
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics …
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.
problem Investigate Higgs bundles and flat connections on compact Sasakian manifolds.
method Introduce quasi-regularity and regularity of vector bundles, relate to orbibundles, extend non-abelian Hodge correspondence.
result Extend non-abelian Hodge correspondence to quasi-regular Sasakian manifolds.
The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.
problem Understanding the topology and geometry of simple orbifolds.
method Generalizing concepts from simple polytopes to simple orbifolds, focusing on simple handlebodies.
result Characterization of orbifold-aspherical properties and the existence of rank-two free abelian subgroups in terms of combinatorics.
We propose a method for determining the spins of BPS states supported on line defects in 4d N=2 theories of class S. Via the 2d-4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface C. Our approach combines the technology of spectral networks…
We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture…