Proves volume cone implies metric cone in nonsmooth spaces.
problem Generalizing a result from smooth spaces to nonsmooth spaces.
method Proves `volume cone implies metric cone' in RCD spaces.
result Generalizes Cheeger-Colding result to nonsmooth spaces.
Calculates volumes of specific cone-manifolds using Schläfli formula.
problem Calculating volumes of specific cone-manifolds.
method Using Schläfli formula.
result Volume of the 732 link cone-manifolds and its cyclic coverings. Proves boundedness of log Fano cone singularities with bounded local volumes.
problem Understanding the boundedness of log Fano cone singularities.
method Analyzes K-semistable log Fano cone singularities with bounded volumes.
result The set of local volumes of klt singularities has zero as the only accumulation point.
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.
Proves stability of cone-volume measure with nearly constant density.
problem Stability of cone-volume measure with near constant density.
method Proves stability of cone-volume measure with near constant density.
result Homothetic copy of the body is close to the unit ball in the L2-distance. We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …
Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…
Study shows expanding Ricci solitons from specific metric cones.
problem Analyzing Ricci flows from weakly PIC1 metric cones.
method Complete weakly PIC1 Ricci flows with Euclidean volume growth.
result Ricci flows must be expanding gradient Ricci solitons.
We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in t…
Exact formulas for volumes of specific knot cone-manifolds.
problem Finding exact volumes of cone-manifolds with two-bridge knots.
method Provided exact integral formulas using Chebyshev polynomials and algebraic equations.
result Exact formulas for hyperbolic and spherical volumes of cone-manifolds.
The paper studies volumes of conformally flat manifolds in light-cone geometry.
problem Volume maximization of conformally flat manifolds in light-cone geometry.
method Computes variational formulas for the volume of hypersurfaces in light-cone.
result Hypersurfaces of conformally flat manifolds maximize volume in certain null hypersurfaces.
Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.
problem Whether the volume growth order of manifolds is greater than or equal to the dimension of their asymptotic cones.
method Analyzing asymptotic cones and volume growth conditions, extending Sormani's results.
result Existence of asymptotic cones with upper box dimension at most equal to the volume growth order.
Study proves Volume Conjecture for Reshetikhin-Turaev invariants.
problem Volume Conjecture for Reshetikhin-Turaev invariants.
method Hyperbolic cone metrics and discrete Fourier transforms.
result Proves Volume Conjecture for most figure-8 knot configurations.
Formulae for volumes of double twist knots found.
problem Calculating volumes of specific hyperbolic cone-manifolds.
method Explicit formulae for volumes of double twist knots.
result Formulae for volumes of double twist knots found.
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…
Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot 41 and the links 512 and 622, have been obtained by the second named author and his collaborators. In this paper we explicitly find the hyperbolic volume for cone-mani…
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…
Calculates volumes of knot cone-manifolds using Schläfli formula.
problem Calculating volumes of specific knot cone-manifolds.
method Schläfli formula, Hilden-Lozano-Montesinos-Amilibia instructions, Ham-Mednykh-Petrov methods, Hoste-Shanahan presentation.
result Concrete and explicit formula for volumes of C(2n,3) cone-manifolds. Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.
New Calabi-Yau metrics found on C^n for n>=3.
problem Finding Calabi-Yau metrics on complex manifolds.
method Constructing metrics with maximal volume growth and singular tangent cones.
result Infinitely many complete Calabi-Yau metrics on C^n.
Study of four-dimensional hyperbolic Dehn filling.
problem Understanding four-dimensional analogues of Thurston's hyperbolic Dehn filling.
method Construction of an analytic path of complete, finite-volume cone four-manifolds interpolating between two hyperbolic four-manifolds.
result Construction of a path of complete, finite-volume cone four-manifolds that interpolates between two hyperbolic four-manifolds.
New proof of a unique 3-part partition in 8D space.
problem Existence of a non-standard isoperimetric partition in high dimensions.
method Analytical proof showing existence of a specific partition.
result Existence of a non-standard isoperimetric partition in 8D space.
We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U-functional of a convex body. For both results we provide stronger versions in the sense of stability i…
Recursion formula derived for moduli spaces of hyperbolic surfaces with cone points.
problem Computing volumes of moduli spaces of hyperbolic surfaces with specific boundary and cone points.
method Using generalized McShane's identities, derived a recursion formula for volumes.
result Obtained a recursion formula for volumes of moduli spaces of hyperbolic surfaces.
Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.
problem Understanding structure of limits of manifolds with Ricci curvature bounds.
method Mosco convergence of Dirichlet energies to Cheeger energy, introduction of monotone quantities, volume convergence.
result Volume convergence to Hausdorff n-measure in limits of manifolds.
We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent …
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, close…
New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
Complete Calabi-Yau metrics found on higher-dimensional complex space.
problem Finding complete Calabi-Yau metrics in higher dimensions.
method Constructing metrics with specific properties on Cn. result Examples of complete Calabi-Yau metrics with Euclidean volume growth.
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
In curved spaces, isoperimetric sets don't exist for small volumes.
problem Nonexistence of isoperimetric sets in spaces of positive curvature.
method Constructing specific noncompact smooth Riemannian manifolds with positive curvature.
result Nonexistence of isoperimetric sets for small volumes in spaces of positive curvature.
Survey on 4-manifolds with specific curvature properties.
problem Understanding the structure of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Analysis of blow-downs and cone-like structures at infinity.
result Manifolds look like cones over spherical space forms at infinity.
The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.
problem Analyzing Ricci flow with Ricci curvature and volume constraints.
method Proving convergence and curvature bounds for Ricci flow with specific constraints.
result Ricci flow with specified constraints converges to a flat cone or static flow.
In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…
Study confirms boundedness of certain singularities in log Fano geometry.
problem Boundedness of log Fano cone singularities and minimal log discrepancies.
method Analyzing local volumes and minimal log discrepancies of Kollár components.
result Boundedness of K-semistable log Fano cone singularities confirmed in dimension three.
Formulae for volumes and A-polynomials of a specific link.
problem Calculating volumes and A-polynomials for a specific type of link.
method Analyzing the double twist link and using character varieties.
result Formulas for the volume and A-polynomial of J(2m+1,2m+1). Study on higher-dimensional quasigeodesics in metric spaces.
problem Understanding asymptotic structure of Morse quasiflats.
method Proving asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity.
result Morse quasiflats exhibit Euclidean volume growth rigidity.
New complete Calabi-Yau metric on C^3 with maximal volume growth.
problem Modeling collapsing Calabi-Yau threefolds near nodal points.
method Existence result perturbed into an actual solution with correction of slowly decaying error terms.
result Complete Calabi-Yau metric on C^3 with maximal volume growth and non-standard geometry near singularity.
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.
problem Bounding harmonic functions on manifolds with specific curvature properties.
method Analyzing the asymptotic volume ratio and eigenvalue counting function.
result Sharp upper bounds for harmonic functions with polynomial growth.
The paper calculates volumes and curvatures in Sasaki geometry using localization formulas.
problem Calculating volumes and curvatures in Sasaki geometry.
method Using Duistermaat-Heckman localization formula and its extensions.
result The Einstein-Hilbert functional attains its minimal value and has at least one Reeb vector field with vanishing transverse Futaki invariant.
The paper proves volume minimization for Kähler-Einstein metrics and related structures.
problem Volume minimization for Kähler-Einstein metrics and related structures.
method Volume minimization using real valuations centered at the vertex of the affine cone.
result The normalized volume is globally minimized at the canonical valuation.
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.