arXiv research

Positive simplicial volume implies locally symmetric space structure.

Optimizes bounds for threefold singularity volumes.

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

The study proves a localization theorem and calculates volumes for superspaces.

The paper proves ACC for local volumes under boundedness conditions.

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, an…

Finite volume ends found in quaternionic Kähler manifolds.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

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…

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite $L^2$ norm in dimension 4.

Characterizes when almost smooth spaces become RCD spaces.

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

Study shows boundedness of klt singularities in 3D or with bounded Kollár components.

Local minimizers are convex and close to Wulff shapes.

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discre…

Hexagonal tilings minimize perimeter with unequal volumes.

Arithmetic spaces' thin parts are negligible, impacting Betti numbers.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Uniform volume estimate for Kähler metrics in big cohomology classes.

We investigate how the local fluctuations of the signed traded volumes affect the dependence of demands between stocks. We analyze the empirical dependence of demands using copulas and show that they are well described by a bivariate $\mathcal{K}$ copula density function. We find that large local fluctuations strongly …

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.

Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.

Study confirms boundedness of certain singularities in log Fano geometry.

Minimal submanifolds in octonionic hyperbolic spaces have large volume.

Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k…

Study finds knots with ideal length need not have smallest volume.

The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equ…

Study shows range of simplicial volumes for open manifolds.

The paper classifies energy-minimizing sets in specific domains.

New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compa…

New families of Ricci solitons found with collapsing volume.

We prove that the locally finite simplicial volume and the Lipschitz simplicial volume are additive with respect to certain gluings of manifolds. In particular, we prove that in dimension $\geq 3$ they are additive with respect to connected sums and gluings along $π_1$-injective, amenable aspherical boundary components…

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian manifold of compact type into its Cartesian products.

Second part of Q-curvature research focusing on volume comparison.

Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…

We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results al…

The article disproves a local systolic inequality and shows a lower bound on filling area.

We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection of hyperbolic 3-manifolds of bounded volume contains infinitely many commensurab…

This paper is mostly a survey of recent work on sequences of locally symmetric spaces whose Riemannian volume goes to infinity. We also work out some applications to random surfaces.