Two unique conic-line arrangements with degree 9 are found.
problem Identifying Zariski pairs of conic-line arrangements.
method Using connected numbers to distinguish topologies.
result Found two pairs of conic-line arrangements with a unique conic.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
The article studies conic connections on complex manifolds and their geometric properties.
problem Characterizing and understanding conic connections on complex manifolds.
method Develops a new approach to the cubic torsion and studies the geometric conditions for its vanishing.
result Provides a geometric condition characterizing the vanishing of cubic torsion.
Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.
problem Modeling conic Laplacian on \(\mb P^1\) with specific boundary conditions.
method Fourier decomposition, Legendre equations, gluing map, Friedrichs spectrum, Weyl function.
result Explicit computation of eigenfunctions and \(S\)-matrix.
Study conic-line arrangements of degree 7, finding their topology and components.
problem Understanding the topology and components of conic-line arrangements of degree 7.
method Examined the embedded topology using SageMath and Sirocco package.
result Determined the number of connected components and fundamental groups.
Study conic line arrangements of degree 7, finding their topology and connected components.
problem Understanding the topology of conic line arrangements of degree 7.
method Identifying a π1-equivalent Zariski pair to prove the existence of a conic line arrangement with specific combinatorics. result Determine the number of connected components of conic line arrangements of degree 7.
Proves Morse index theorem for geodesics in conic Finsler manifolds.
problem Geodesic index theorem in conic Finsler manifolds with variable endpoints.
method Proves Morse index theorem for geodesics connecting submanifolds in a C7 manifold with a C6 conic pseudo-Finsler metric. result Establishes the Morse index theorem for geodesics in conic Finsler manifolds.
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical sur…
Paper proves Whitney stratified spaces can be given a conically smooth structure.
problem Proving Whitney stratified spaces can be given a conically smooth structure.
method Introduced conically smooth structure by Ayala, Francis, and Tanaka. Proved conjecture that any Whitney stratified space admits a canonical conically smooth structure.
result Established a connection between Whitney stratified spaces and conically smooth spaces.
We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local braid monodromies and the techniques of Moishezon-Teicher and van-Kampen. We al…
We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with c…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
As an application of the method of [4], we find the metric and connection on the space of conics in CP2 determined as the solution space of the ODE eqn(1). These calculations underpin the twistor construction of the Radon transform on conics in CP2 described in [5]. Two further examples of the m…
Study on spherical conical metrics and their reducibility on compact Riemann surfaces.
problem Existence and geometric structure of reducible spherical conical metrics.
method Analysis of monodromy groups and geometric cutting of surfaces.
result Existence of reducible spherical conical metrics with saddle points on the same geodesic.
We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of …
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum G2 manifolds. The paper studies the topology of spherical tori with one conical point.
problem Determining the topology of surfaces with constant curvature and conical points.
method Analyzing the moduli space of genus one surfaces with a conical point of specific angles.
result The moduli space topology depends on the integer m>0 for ϑ∈(2m−1,2m+1), and it has a complex structure for ϑ=2m. A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…
Joachimsthal integrals characterize conics in various geometries.
problem Characterizing conics in different geometries using Joachimsthal integrals.
method Extending Joachimsthal integrals to spherical and hyperbolic geometries and connecting them to the Poritsky property.
result Existence of Joachimsthal integrals characterizes conics in various geometries.
We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.
The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
Branched covers between Riemann surfaces are associated with certain combinatorial data, and Hurwitz existence problem asks whether given data satisfying those combinatorial constraints can be realized by some branched cover. We connect recent development in spherical conic metrics to this old problem, and give a new m…
We give an example of a homogeneous reflexive sheaf over C3 which admits a non-conical Hermitian Yang-Mills connection. This is expected to model bubbling phenomenon along complex codimension 2 submanifolds when the Fueter section takes zero value.
Mathematical analysis shows Delisle-Euler map methods are optimal.
problem Comparing ancient and modern map drawing methods.
method Analyzing similarities and differences between ancient and modern map drawing methods.
result Delisle-Euler map methods are optimal among conical maps.
Dunkl connections on complex plane don't preserve metrics.
problem Preserving metrics with Dunkl connections on \(\mathbb{C}^2\).
method Analysis of the topology of spherical tori with conical points.
result General Dunkl connections on \(\mathbb{C}^2\) do not preserve non-zero Hermitian forms.
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter…
Let M be a manifold and Λ a compact exact connected Lagrangian submanifold of T∗M. We can associate with Λ a conic Lagrangian submanifold Λ′ of T∗(M×R). We prove that there exists a canonical sheaf F on M×R whose microsupport is Λ′ outside the zero section. We deduce the already known re…
The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.
problem Investigating the asymptotic behavior of G2-monopoles on nonparabolic G2-manifolds.
method Proof of finite mass for monopoles with bounded curvature on nonparabolic G2-manifolds; sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
result Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the r…
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
Formula derived for spectral determinant of sphere with conical singularities.
problem Calculating the spectral determinant of a sphere with conical singularities.
method Explicit closed formula derived using zeta regularization and Liouville action.
result Metrics with equal conical angles are a stationary point of the determinant, and a minimum if surface area is small.
Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
We develop the deformation theory of instantons on asymptotically conical G2-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted Dirac operator on the G2-manifold and to the eigenvalues of a twisted Dirac op…
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…
The paper classifies isoparametric hypersurfaces in conic Finsler spaces.
problem Identifying new isoparametric hypersurfaces in conic Finsler spaces.
method Introduced isoparametric functions and hypersurfaces in conic Finsler spaces, classified them in specific spaces.
result Found additional isoparametric hypersurfaces in conic Minkowski spaces, such as helicoids.
New gradient Ricci solitons found for SU(2) invariants.
problem Exploring gradient Ricci solitons with SU(2) symmetry. method Construction of new solitons using cohomogeneity one group actions.
result 3-parameter families of complete SU(2)-invariant asymptotically conical expanding gradient Ricci solitons. Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
Proves conditions for positive scalar curvature on certain manifolds with conical singularities.
problem Conditions for positive scalar curvature on manifolds with isolated conical singularities.
method Analyzes isolated conical singularities and uses Geroch type results.
result No metric with positive scalar curvature on X#Tn with isolated conical singularity. Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family…
We consider the Chern connection of a (conic) pseudo-Finsler manifold (M,L) as a linear connection ∇V on any open subset Ω⊂M associated to any vector field V on Ω which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor g.…