arXiv research

Paper solves Minkowski problem for q-torsional rigidity using curvature flow.

Paper solves Minkowski problem for anisotropic p-torsional rigidity.

Paper solves Minkowski problem for k-torsional rigidity.

The paper examines torsional rigidity bounds under geometric flows.

New method solves generalized Minkowski problem for torsional rigidity.

This paper solves the dual Minkowski problem for q-torsional rigidity.

Paper introduces new $L_p$ $q$-torsional measure and solves Minkowski problem.

Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsi…

We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs…

Projective structures are mostly rigid at the boundary but some are not.

Researchers find limits on curvature of certain 3D solitons.

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\niγ$-torsional rigidity $\mathcal{T}_{γ,\mathsf{g}}$ on a complete Riemannian two-manifold $(\mathbb M^2,\mathsf{g})$. Even in the special case of $\mathbb R^2$, major results …

Rigidity proven for a specific type of solitons with harmonic curvature.

Paper solves a new Minkowski problem for a specific type of rigidity.

This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle

Paper studies rigidity of translation surfaces in 3D sphere using quaternionic product.

Graphically discrete groups have strong rigidity properties.

In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.

Study graph products of groups, classifying them up to measure equivalence and rigidity.

Study on deformations of LC Spin(7) instantons simplifies the problem.

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…

Groups with certain properties have invariant subalgebra rigidity.

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley…

Flow solves $G_2$ system, proving existence of torsion-free metrics.

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form…

Study rigidity of PSU(1,1) actions on circle via harmonic measures.

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form $G\rtimes\mathbb Z$. If in addition $Wh(G\times \mathbb Z)=0$, which is true for …

Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,Ω\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (Ω)\,$ (known as {\em torsional rigidity} associated to $Ω$) among all domains of prescribed volume $v$. Our results show tha…

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we…

Study on scalar curvature deformations in pseudohermitian manifolds.

Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.

We show that for a closed $n$-manifold $N$ admitting a quasiregular mapping from the Euclidean $n$-space the following are equivalent: (1) order of growth of $π_1(N)$ is $n$, (2) $N$ is aspherical, and (3) $π_1(N)$ is virtually $\mathbb{Z}^n$ and torsion free.

The paper proves rigidity for cocycles from higher rank lattices to Out(FN).

New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

This article announces joint work with Frank Connolly and Jim Davis. We generalize our classification of pseudo-free involutions on the n-torus, by studying the action of the associated infinite group with torsion in the universal cover. Included is a non-Riemannian example obtained from the restriction of the action o…

Model proteins with bonds using Kauffman bracket skein module.

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with …

Let $Γ$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\geq2$. We introduce the notion of volume for a representation $ρ:Γ\rightarrow PU(m,1)$ where $m \geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $ρ_n:…

Study on compact strong HKT manifolds and their properties.

Quantization and reduction studied for CR manifolds with group actions.

We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection motivates new versions of the volume and Laplacian comparison theorems that are valid…

Effective rank rigidity proved for cubulated groups with factor systems.

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups $\Out(G)$ is infinite. We a…