We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…
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48 results for “conic 2-spheres”
Sharp volume bounds for conic 2-spheres with Gaussian curvature constraints.
problem Volume constraints for conic 2-spheres with curvature bounds.
method Level set analysis and iso-perimetric inequality with new analytical tools.
result Computed minimal volume of conic spheres with curvature bounded by 1.
A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.
New condition found for spherical metrics with conical singularities.
problem Existence of spherical metrics with conical singularities on 2-spheres.
method Sufficient and necessary conditions derived using existing results.
result Necessary condition for spherical metrics with conical singularities.
New approach to prescribing Gaussian curvature on spheres with conical singularities.
problem Prescribing Gaussian curvature on the 2-sphere with conical singularities.
method Variational methods not relying on Moser-Trudinger inequality, plus precompactness theorem.
result Sufficient conditions for a positive function to be the Gaussian curvature of a conformal conical metric.
In this article we give a criterion for the existence of a metric of curvature on a -sphere with conical singularities of prescribed angles and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…
Hyperbolic 2-spheres with conical singularities, accessory parameters and Kaehler metrics on math.CV
We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic 2-sphere with conical singularities of arbitrary orders , generates accessory parameters of the as…
The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow …
This paper solves part of a problem by constructing surfaces with specific curvature and singularities.
problem Solving an open problem by Gálvez, Hauswirth, and Mira regarding constant curvature metrics with conical singularities.
method Established a geometric correspondence between metrics and isometric immersions into Euclidean 3-space, constructing a family of surfaces.
result Explicitly constructed a family of surfaces with constant curvature one and two conical singularities.
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 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.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves …
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.
CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 …
New measures found in 3-uniform geometry.
problem Understanding non-flat uniform measures in geometric measure theory.
method Combining combinatorial methods and distance symmetry properties.
result Infinite family of 3-uniform measures constructed.
A conformal metric with constant curvature one and finite conical singularities on a compact Riemann surface can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function on , called the {\it developing …
New method proves Alexandrov theorem for curved spacetimes.
problem Proving Alexandrov theorem for curved surfaces with conical singularities.
method Adapting Volkov's variational method to handle Lorentzian angles.
result Existence of a locally Minkowski 3-manifold with conical singularities isometric to a given surface.
In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called…
Unique hyperbolic affine spheres associated with convex cones in 3D.
problem Characterizing and classifying self-associated convex cones in 3D.
method Using affine spheres, hyperbolic geometry, and differential equations.
result Complete classification of self-associated convex cones and their associated affine spheres.
The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of -vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components.
result The subgraph of -vertex flag 2-spheres is connected, while the subgraph of -vertex stacked 2-spheres has at least as many connected components as trees with specific properties.
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
Paper generalizes convexity to conic Mabuchi's functional.
problem Generalizing convexity to conic settings.
method Established frame for conic cscK metrics, introduced conic Mabuchi functional, proved convexity along geodesics.
result Proved convexity of conic Mabuchi's functional.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
Sharp mass bounds for ALE and ALF toric 4-manifolds.
problem Establishing lower bounds for the mass of ALE and ALF toric 4-manifolds.
method Using gravitational instantons and conical angle defects, the mass is bounded below by a sum of the mass of the corresponding instanton and an expression determined by conical angle defects.
result The mass of an ALE or ALF toric 4-manifold is not less than the mass of the corresponding gravitational instanton.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
The paper studies minimal 2-spheres of constant curvature in complex hyperquadrics using matrix theory.
problem Exploring minimal 2-spheres of constant curvature in complex hyperquadrics.
method Using singular-value decomposition of complex matrices to study the moduli space of noncongruent spheres.
result Uniqueness proven for totally real constantly curved 2-spheres.
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.
Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.
problem Obstructing homotopic embeddings of 2-spheres into 5-manifolds from being isotopic.
method Using a level preserving Whitney move in codimension 3 to eliminate double points, and classical methods.
result New results for simply-connected 5-manifolds and 2-spheres with algebraic dual 3-spheres.
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
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.
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.
The degree of certain holomorphic 2-spheres is bounded.
problem Understanding the degree of holomorphic 2-spheres in complex Grassmannians.
method Analyzing the degree of linearly full constantly curved holomorphic 2-spheres in G(2,n+2;C).
result The degree is bounded between n and 2n for specific n values.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space for and .
result Detailed description of spaces for and examples for .
The abstract discusses maximal rank 4-webs formed by confocal conics and their properties.
problem Characterizing confocal conics using 4-webs of maximal rank.
method Examining three different 4-webs formed by confocal conics and proving they are of maximal rank.
result Confocal conics are characterized from the web theory viewpoint.
Smooth conic Kähler metrics with uniform curvature bound.
problem Constructing smooth conic Kähler metrics with bounded curvature.
method Using upper bisectional curvature bound and choosing a new background metric.
result Constructing a smoothing sequence for conic metrics.
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.
Modified flow smooths conical Kähler-Ricci problem.
problem Smooth approximation of modified conical Kähler-Ricci flow.
method Constructing a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
result Long-time solution of the modified conical Kähler-Ricci flow.
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 with isolated conical singularity.
Researchers find necessary conditions for conic 4-spheres with constant σ₂ metrics.
problem Finding constant σ₂ metrics on conic 4-spheres.
method Analyzing earlier works and using necessary conditions to find existence.
result Uniqueness of conic 4-spheres with constant σ₂ metrics under certain conditions.
Proves conic version of YTD conjecture on log Fano manifolds.
problem Existence of conic Kahler-Einstein metrics on log Fano manifolds.
method Proof of the YTD conjecture for conic Kahler-Einstein metrics.
result Proven existence of conic Kahler-Einstein metrics on log Fano manifolds.
Study Schauder estimates for conical singularities using Kähler metrics.
problem Derive interior Schauder estimates for equations with conical singularities.
method Use linear elliptic and parabolic equations with a Kähler metric of conical singularities.
result Prove short-time existence of conical Kähler-Ricci flow.
In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than ; in particular, we define and study the Teichmüller space of conic constant curvature metrics on a surface of genus with …
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
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.
Study constant Q-curvature metrics on conic 4-manifolds.
problem Find metrics with constant Q-curvature on conic 4-manifolds.
method Analyze related differential equations in the given conformal class.
result Solve constant Q-curvature problem on conic 4-manifolds.
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.