The paper calculates delta invariants for specific geometric structures.
problem Computing delta invariants for projective bundles and cones of Fano type.
method Provides a precise formula for delta invariants.
result A formula to compute delta invariants for projective bundles and cones of Fano type.
Newly confirmed area-minimizing properties of Lawson-Osserman cones.
problem Verifying the area-minimizing property of Lawson-Osserman cones.
method Analyzing cones of type (n, p, 2) constructed in [XYZ].
result All Lawson-Osserman cones of type (n, p, 2) are area-minimizing.
Paper constructs flows converging to cones and foliations.
problem Understanding mean curvature flow convergence to cones and foliations.
method Constructs a family of mean curvature flows converging to cones and foliations under specific conditions.
result Flow converges to area minimizing, strictly stable hypercone and Hardt-Simon foliation of the cone.
The study explores the types of Sasakian structures within a Sasaki cone and their properties.
problem Understanding the types of Sasakian structures within a Sasaki cone.
method Analyzing the Sasaki cone's dimension and properties of Sasakian structures.
result The types of Sasakian structures within a Sasaki cone can be either all positive, all indefinite, or a mix of both.
We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, th…
New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.
problem Uncertainty in homeomorphism types of tangent cones of non-collapsed Ricci limit spaces.
method Construction of limit spaces in all dimensions at least 5.
result Any finite collection of manifolds can appear as cross sections of tangent cones of the same point.
Study transverse measures on infinite type hyperbolic surfaces.
problem Characterize the cone of transverse measures on infinite type hyperbolic surfaces.
method Use inverse limits and geodesic laminations to describe and construct cones of transverse measures.
result Explicit descriptions and bases of cones of transverse measures exist for many laminations.
Study shows instability of specific cone solutions in high-dimensional spaces.
problem Unstable solutions of minimal graphs in high codimension.
method Min-max technique applied to Euclidean spaces.
result First examples of non-smooth unstable minimal graphs.
I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…
The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.
problem Characterizing constant mean curvature surfaces in a three-dimensional light cone.
method Analyzing entire constant mean curvature graphs in the light cone Q+3 under the condition of bounded Gaussian curvature. result Entire constant mean curvature graphs in the light cone are either horospheres or spheres.
Authors characterize Lagrangian cone structures from (2,3,5)-distributions.
problem Characterizing Lagrangian cone structures from (2,3,5)-distributions. method Characterization via pseudo-product structures of type G2. result Completion of duality between (2,3,5)-distributions and Lagrangian cone structures. Study on minimizing singular capillary cones with stability and instability results.
problem Minimizing singular capillary cones with free boundary.
method Stability criterion à la Jerison-Savin, Simons-type inequality for convex, homogeneous, symmetric functions of principal curvatures, boundary condition specific to capillary setting.
result Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4, and non-trivial axially symmetric cones are unstable in dimensions up to 6.
Study examines causal properties of Finsler spacetimes with cone Killing vectors.
problem Characterize causality in Finsler spacetimes with specific Killing vectors.
method Explores the relationship between wind Riemannian structures and spacetimes with cone Killing vectors, focusing on Finsler-Kropina metrics.
result Characterizes causality properties using metric-type properties of Finslerian structures.
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
Paper proves minimal surfaces near quadratic cones have specific smooth structure.
problem Characterize minimal surfaces near quadratic cones.
method Analyzes n-varifolds in the unit ball close to a minimizing quadratic cone. result Singularities modeled on these cones determine the local structure of nearby minimal surfaces.
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.
We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2π, i.e. contained in the interval (0,2π). In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a…
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…
New insights into Kähler Ricci solitons and Calabi-Yau cones.
problem Understanding Kähler Ricci solitons and their relationship to Calabi-Yau cones.
method Analyzing the canonical cone of Fano manifolds and using openness of weight functions.
result The canonical cone of a product of a smooth Fano manifold and a complex projective space is a Calabi-Yau cone under certain conditions.
Paper shows examples of hyperbolic cone structures degenerating with decreasing cone angles.
problem Degeneration of hyperbolic cone structures with specific cone angles.
method Constructed examples of hyperbolic cone structures on a certain alternating link in the thickened torus.
result Example of degeneration of hyperbolic cone structures with decreasing cone angles less than 2π.
The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.
We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the ge…
The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
Proves uniqueness of tangent cones for specific types of connections.
problem Uniqueness of tangent cones for Hermitian Yang-Mills connections with isolated singularities.
method Simple direct proof using μ-polystable holomorphic bundles over P^n-1.
result Uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections.
Researchers compute the index of a specific operator on contact manifolds.
problem Computing the index of a twisted Dolbeault operator on toric contact manifolds.
method Using equivariant techniques, they localized the symbol to Reeb orbits and applied polytope decomposition.
result They derived an Atiyah-Bott-Lefschetz type formula for the index.
Study proves radial symmetry in convex cones using subharmonic functions.
problem Proving radial symmetry in convex cones with boundary conditions.
method Using maximum principle and integral identities for subharmonic functions.
result Proves radial symmetry and Serrin-type results for partially overdetermined problems.
We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometr…
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.
In this paper we investigate the singularities of Lagrangian mean curvature flows in Cm by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…
The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.
problem Stochastic invariance of cones in SPDEs with jumps.
method Sufficient conditions for stochastic invariance of closed convex cones in abstract L2-spaces. result Conditions for stochastic invariance of cones are provided and analyzed.
We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extre…
Paper approximates Kähler metrics with cone singularities near a hypersurface.
problem Approximating Kähler metrics near a hypersurface with cone singularities.
method Using conical approximations and holomorphic vector fields, the paper shows how to approximate Kähler metrics of Poincaré type near a smooth hypersurface.
result Constant scalar curvature Kähler metrics can be approximated by those with cone singularities of small angle along a hypersurface.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
Symplectic vortex equations link Sasakian manifolds to Kahler cones.
problem Existence and uniqueness of symplectic vortex solutions.
method Hitchin-Kobayashi correspondence and Kazdan-Warner equations.
result Construction of a map between vortex solutions and effective divisors.
New method connects compression bodies through cone manifolds.
problem Understanding how compression bodies can be transformed.
method Using cone manifold holonomy groups and standard CAT(0) space techniques. result Realized all edges in compression body graph through paths.
Study proves uniqueness of hyperbolic cone structures up to isotopy.
problem Determining hyperbolic cone structures on surfaces up to isotopy.
method Analyzes geodesic lengths of specific homotopy classes of curves.
result Thurston metric is well-defined on Teichmüller space of hyperbolic cone surfaces.
Study on extremals in sub-Lorentzian geometry defined by antinorm.
problem Characterizing extremals in sub-Lorentzian structures.
method Deriving Hamiltonian system and conditions for extremal trajectories.
result Conditions for normal extremal trajectories and properties of abnormal extremals.
Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.
problem Serrin's overdetermined problem in convex cones of Riemannian manifolds.
method Rigidity results, soap bubble theorem, Heintze-Karcher inequality, drift Laplacian analysis.
result Characterization of intersections of geodesic balls with cones in Riemannian manifolds.
Study of geometric structures on projective space complement without Schwarz conditions.
problem Geometric structures on projective space complement without Schwarz conditions.
method Use of Dunkl system to study geometric structures.
result Space is a cone-manifold.
A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the con…
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
Recent progress on minimal surface system and cones in Euclidean spaces.
problem Exploring the Dirichlet problem for minimal surfaces and cones.
method Systematic developments and new families of minimizing cones.
result New families of minimizing cones of different types.
Proof shows cones minimize certain geometric functionals.
problem Minimizing cones over spheres in geometric functionals.
method Proof by foliation analysis of cone leaves.
result Cone minimizes functionals for SkimesSl. The paper explores transformations between power law problems and geodesics on cones.
problem Solving power law problems and understanding their geometric properties.
method Geometric transformations and cone metrics.
result Derivation of Maclaurin duality and Jacobi-Maupertuis metric reformulation.
Study vanishes L2 cohomology groups on Hessian manifolds and cones.
problem Vanishing L2 cohomology groups on Hessian manifolds and cones. method Analyzes L2 cohomology groups of Kodaira-Nakano type on complete Hessian manifolds and a regular convex cone with the Cheng-Yau metric. result Obtains vanishing theorems for L2 cohomology groups on Hessian manifolds and cones. Study geometric criteria for cone structures on flag varieties.
problem Geometric criteria for isotrivial cone structures on flag varieties.
method Complete classification of flag varieties satisfying a projective-geometric criterion.
result Characterization of flag varieties with an injective Gaussian map.
The paper studies conical structures and Perelman's functionals on Ricci flows.
problem Characterizing conical structures and Perelman's functionals on manifolds.
method Analysis of Perelman's functionals on cones and adaptation of the pseudolocality theorem.
result Cone structures can be smoothed out by type III immortal solutions on Ricci flows.
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.