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.
Study examines heart and football-shaped metrics, verifying geometric structure.
problem Analyzing reducible spherical conical metrics and their geometric properties.
method Examined 1-parameter heart shape and 3-parameter football shape families, verified structure theorem, used explicit metric and geodesic calculations.
result Naturally arise from Abelian differentials of the third kind, offer new evidence for spherical geometry and complex analytic structure interaction.
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 …
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
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.
The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.
problem Existence and properties of eigenfunctions for spherical conical metrics.
method Application of multivalued harmonic maps to spheres and algebraic constructions.
result New criteria and examples of metrics with many 2-eigenfunctions.
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…
We study the deformation of spherical conical metrics with at least some of the cone angles larger than 2π. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction c…
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
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…
Suppose that θ1,θ2,…,θn are positive numbers and n≥3. Does there exist a sphere with a spherical metric with n conical singularities of angles 2πθ1,2πθ2,…,2πθn? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when w…
We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with s…
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.
Study spherical metrics on flat torus with cone singularities.
problem Existence and uniqueness of spherical metrics with specific cone angles.
method Analysis of Green function critical points and torus geometry.
result Existence of unique spherical metrics determined by torus geometry.
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. Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ1,…,2πϑn and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…
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. We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribe…
A simple proof is given of the necessary and sufficient condition on a triple of positive numbers A,B,C for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles A,B,C. The same condition is necessary and sufficient for the triple A,B,C to be in…
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
The paper proves formulas for determinant determinants of Laplacians on Riemann surfaces with conical singularities.
problem Determinants of Laplacians on Riemann surfaces with conical singularities.
method Polyakov-Alvarez type comparison formulas for determinants of Friedrichs extensions of Laplacians.
result Determine how determinants depend on conical singularities and provide explicit formulas.
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.
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.
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.
The paper studies T-tensor of spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition.
problem Characterizing spherically symmetric Finsler metrics with vanishing T-tensor.
method Deriving a general expression for the T-tensor and characterizing metrics satisfying the T-condition.
result Characterization of spherically symmetric Finsler metrics with vanishing T-tensor.
Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.
problem Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
method Applied Perron's method and Thurston's algorithm to prove existence and convergence.
result Existence and uniqueness of circle patterns on surfaces with prescribed geodesic curvatures.
3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.
This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.
problem Extension of quadric surfaces of revolution to 3-sphere.
method Rigorous classification and characterization using spherical angular momentum.
result Spherical ellipsoids, hyperboloids, and paraboloids are Weingarten surfaces with a specific cubic relation between principal curvatures.
Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.
problem Exploring various anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Presented various anisotropic conformal transformations including C-anisotropic, horizontal C-anisotropic, and vertical C-anisotropic transformations. result Vertical φT-condition transformation makes every Landsberg surface Berwaldian. New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.
problem Existence and classification of conical Calabi-Yau structures on horospherical cones.
method Variational approach to establish equivalence between volume minimization and existence of conical Calabi-Yau structures.
result Existence of many irregular horospherical cones with mild singularities.
Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.
problem Existence and non-uniqueness of cone spherical metrics with prescribed singularities.
method Utilizing polystable extensions of line bundles, the study establishes three primary results concerning these metrics.
result Existence of multiple irreducible and reducible cone spherical metrics for certain effective divisors.
Quiver varieties' geometry at infinity studied using Nakajima metric.
problem Understanding the geometry at infinity of quiver varieties.
method Using Melrose's approach to study the geometry at infinity of the Nakajima metric on reduced Hilbert schemes.
result Quiver varieties are quasi-asymptotically conical under generic conditions.
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.
Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.
problem Understanding the topology of conic-line arrangements using Poncelet's theorem.
method Study unramified double covers induced by Poncelet transverses.
result Existence of families of Zariski pairs of degree 2m+6 for m≥2. A conformal metric g with constant curvature one and finite conical singularities on a compact Riemann surface Σ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function f on Σ\{singularities}, called the {\it developing …
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.
Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.
problem Proving a generalized Gauss-Bonnet formula for conical metrics.
method Analyzing Gaussian curvature and Lebesgue integrability.
result Proved a generalized Gauss-Bonnet formula for conical metrics.
We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum', as well as the `off-spectrum', spectral family is Fredholm in function spaces…
The space of Lamé functions is mapped to a Riemann surface with known topology.
problem Understanding the structure of the space of Lamé functions and its relation to Abelian integrals.
method Isomorphic mapping to elliptic curves and Abelian differentials with specific properties.
result The space of Lamé functions is a Riemann surface of finite type with known genus and Euler characteristic.
The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.
problem Classifying Landsberg metrics on a 2D Lie group.
method Analyzing left invariant conic Finsler metrics on a 2D non-Abelian Lie group.
result Any left invariant conic Landsberg metric on G must be Berwald. Paper provides an alternative proof for Dahmen's conjectures about Lame equations.
problem Proving the conjecture about the number of Lame equations with finite monodromy.
method Applying results on spherical tori with one conical singularity to Lame equations.
result Alternative proof of Dahmen's conjectures.