arXiv research

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

Graphs with stronger curvature grow faster.

This paper proves a rigidity result for annuli in $RCD(K, N)$-spaces.

New method to find surfaces in null cones with constant curvature near black hole indicators.

Proves boundedness of log Fano cone singularities with bounded local volumes.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

Proves stability of cone-volume measure with nearly constant density.

We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.

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.

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.

We calculate the volume of the $7_3^2$ link cone-manifolds using the Schläfli formula. As an application, we give the volume of the cyclic coverings over the link.

In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove …

The paper studies volumes of conformally flat manifolds in light-cone geometry.

Let $φ\in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $φ$ determines a free-by-cyclic group $Γ=F_n \rtimes_φ\mathbb Z,$ and a homomorphism $α\in H^1(Γ; \mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovi…

Study proves Volume Conjecture for Reshetikhin-Turaev invariants.

Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $…

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 $4_1$ and the links $5^2_1$ and $6^2_2$, 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…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

Study shortest geodesics on flat cone spheres with conical singularities.

New Calabi-Yau metrics found on complex symmetric spaces.

We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's diameter, we argue that the method performs well when initialized with a second-…

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.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

Formula for transgressions on polyhedral manifolds, linking face volumes and outer angles.

New proof of a unique 3-part 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 $\mathrm{U}$-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.

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 …

Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.

We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbo…

The volume conjecture is extended for surface diffeomorphisms with quantum invariants.

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…

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_γ \frac{1}{e^{\ell(γ)}+1}={1/2} \] where $γ$ varies over the homotopy classes of essential simple closed curves and $\ell(γ)$ is the length of the geodesic representative of $γ$. We prove tha…

New examples of Calabi-Yau 3-folds with unique properties.

Outer billiards maps on foliated surfaces with specific vector fields.

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbo…

Active Learning (AL) is a learning task that requires learners interactively query the labels of the sampled unlabeled instances to minimize the training outputs with human supervisions. In theoretical study, learners approximate the version space which covers all possible classification hypothesis into a bounded conve…

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.

Survey on 4-manifolds with specific curvature properties.

In curved spaces, isoperimetric sets don't exist for small volumes.