arXiv research

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

Study shows simplicial volume of certain fiber bundles is zero.

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvatu…

Sharp inequality in spaces with non-negative Ricci curvature.

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

We compute the space of $L^2$ harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

The study shows how certain surfaces can be filled by hyperbolic manifolds.

Linear bound on Betti numbers of negatively curved orbifolds.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

We prove that a reversible Berwaldian Finsler structure $\left(M,F\right)$ with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

Non-Negative Matrix Factorization, NMF, attempts to find a number of archetypal response profiles, or parts, such that any sample profile in the dataset can be approximated by a close profile among these archetypes or a linear combination of these profiles. The non-negativity constraint is imposed while estimating arch…

It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in Kähler manifolds of negative Ricci curvature. The only known result in this direction is a theo…

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

The paper proves conjectures and classifies metrics on 3D manifolds.

Extends conformal prediction to contrastive learning for better coverage of positive samples.

We compute the $L^p$-cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds.

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This …

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

Study shows negatively curved manifolds' spherical volume equals minimal surface area.

We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two…

Study on entropy stability in product spaces of negatively curved symmetric spaces.

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

We show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we a…

Lower bound for Steklov eigenvalues on negatively curved manifolds.

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, the…

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $Γ$. If the quotient manifold $\bar X= Γ\backslash X$ is asymptotically $1/4$-pinched, we prove that $Γ$ is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is t…

We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of $M$ vanishes if $M$ is geometrically finite. Furthermore, when $M$ is a $\mathbb{R}$-rank one locally symmetric space, we show that the bou…

In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature …

Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.

Smooth surface encloses less volume than a ball.

We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we…

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.