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.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…
A new matrix factorization method that approximates data without requiring nonnegativity or convexity.
problem Approximating data matrices without the constraints of nonnegativity or convexity.
method A multi-objective optimization problem finds conical combinations of templates that approximate a given data matrix.
result The method allows for approximation of data sets without the usual constraints of nonnegativity or convexity.
Proves mean curvature flow from conical singularities to shrinkers.
problem Proving mean curvature flow from conical singularities to shrinkers.
method Ważewski box argument
result Existence of embedded closed hypersurface with mean curvature flow.
New conical metrics found on toric varieties with convex cones.
problem Finding conical metrics on toric affine varieties.
method Existence result for inhomogeneous Monge-Ampere equation, transversal a priori estimates.
result Existence of conical Ricci flat Kahler metrics on Q-Gorenstein affine toric varieties.
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 continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regulari…
Overlay framework simplifies exotic derivative pricing.
problem Inconsistent pricing models for exotics.
method Combines path reweighting and conic optimisation.
result Practically model-independent price bands for exotics.
The paper defines conic reach and shows polynomial parallel volume in the plane.
problem Understanding geometric properties of sets in the plane.
method Introducing conic reach and using local Steiner formula to show polynomial volume.
result Polynomial parallel volume of sets in the plane with conic reach.
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.
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 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.
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 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. 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 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 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 2π; in particular, we define and study the Teichmüller space Tγ,kconic of conic constant curvature metrics on a surface of genus γ with k …
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…
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.
Study describes ALF instantons with conical singularities.
problem Understanding ALF instantons with conical singularities.
method Applied techniques from previous work.
result Only 4D subfamilies can be smoothly compactified with conical singularities.
Study X-ray transform on conic spaces, proving injectivity under certain conditions.
problem Injectivity of geodesic X-ray transform on conic metrics.
method Injectivity under non-trapping and no conjugate point assumptions.
result Injectivity of geodesic X-ray transform for asymptotically conic metrics.
We establish a parabolic version of Tian's C2,α-estimate for conical complex Monge-Ampere equations, which includes conical Kähler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler-Ricci flow in arXiv:1411.7284.
We prove that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time t∈[0,+∞). These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C1,β is negative or zero, the corresponding conical Kähler-Ricci flows co…
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…
The paper discusses a twisted conical Kähler-Ricci flow and its convergence properties.
problem Diameter bound and Gromov-Hausdorff convergence of a twisted conical Kähler-Ricci flow.
method Direct arguments on the conical equation without smooth approximation.
result The conical Kähler-Ricci flow will instantly have bounded scalar curvature for t>0. Automates model selection for GLMs using optimization.
problem Automating model selection for generalized linear models.
method Mixed-integer conic optimization for feature subset selection and model constraints.
result Optimization of AIC and BIC criteria with multicollinearity constraints.
The study proves convergence of conic 4-spheres' geometry to boundary cases.
problem Convergence of conic 4-spheres' geometry to boundary cases.
method Proved a convergence theorem on the moduli space of constant σ₂ metrics for conic 4-spheres.
result When a numerical condition converges to the boundary case, conic 4-spheres' geometry converges to the boundary case while preserving capacity.
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.
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.
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.
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. Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
Paper proves unique conical tangent flows in all dimensions.
problem Proving uniqueness of conical singularities in mean curvature flows.
method Based on Chodosh-Schulze's work in hypersurfaces, extended to all codimensions.
result Uniqueness of asymptotically conical tangent flows in all codimensions.
Generically, topological insulators have conical points leading to Dirac-like currents.
problem Understanding the conical structure of degeneracies in topological phases of matter.
method Analyzing Hermitian matrices with three parameters to show conical points.
result Adiabatic deformations of topological insulators result in Dirac-like currents whose total conductivity equals the chiral number of conical points.
The Yamabe flow preserves conical singularities under certain conditions.
problem Preserving conical singularities under the Yamabe flow.
method Using maximal Lq-regularity theory for conically degenerate operators. result Asymptotic expansion of the evolving metric near conical tips.
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
FS&P uses birth-death process to ensure global convergence of stochastic conic particle gradient descent.
problem Global optimization of non-convex objective functions over measure space.
method Introduces Fast Spawn\&Prune (FS\&P) combining CPGD with birth-death process.
result First theoretical guarantee of global convergence for discrete-time stochastic algorithms.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
Proves positive mass theorem on conical manifolds with small angles.
problem Proving the positive mass theorem on conical manifolds with small cone angles.
method Analyzes conical manifolds with small cone angles, assuming spin structure and locally conformal flatness.
result Proves the positive mass theorem under specified conditions.
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. 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.