arXiv research

Paper solves Hessian equations on Kähler manifolds.

New divergence identity for scalar curvature helps prove rigidity of tensors.

The paper proves an inequality for symmetric polynomials under a fixed point measure.

Extends double linear policy with time-varying weights and proves robust positive expectation.

We give elementary constructions for Satake-Furstenberg, Martin and Karpelevich boundaries of symmetric spaces. We also consruct some "new" boundaries

Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the ho…

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earl…

Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.

Proves existence of smooth metrics with specific curvature properties.

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $σ_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n…

We investigate a special kind of contraction of symmetric spaces (respectively, of Lie triple systems), called homotopy. In this first part of a series of two papers we construct such contractions for classical symmetric spaces in an elementary way by using associative algebras with several involutions. This constructi…

The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentu…

The paper proves curvature estimates for specific hypersurfaces in hyperbolic space.

Study solves complex Hessian equation on Hermitian manifolds.

In this paper, we study the problem of conformally deforming a metric on a $3$-dimensional manifold $M^3$ such that its $k$-curvature equals to a prescribed function, where the $k$-curvature is defined by the $k$-th elementary symmetric function of the eigenvalues of the Einstein tensor, $1\le k\le 3$. We prove the sol…

Study quotients of curve complex actions by mapping class group.

We give an elementary (not cut just paste) proof of results of Bott and Shchepin: the space of non-empty subsets of a circle of cardinality at most 3, which is called the third symmetric potency of the circle, is homeomorphic to a 3-sphere and the inclusion of the space of one element subsets is a trefoil knot. Moreove…

The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors"…

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable a…

In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.

We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characte…

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of exi…

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov $n$-algebras and Jord…

The Hecke algebra H_n contains well known idempotents E_λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_λ was given by Aiston and Morton. The closure of E_λ becomes an element Q_λ of the skein of the annulus. In this skein, they are known…

Simplified proof of Gaussian concentration inequality using covariance.

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$ by $σ_k^α$, where $σ_k$ is the $k$-th elementary symmetric function of the principal curvatures and $α\ge 1/k$. We prove that for any $n\geq3$ and any fixe…

New capillary Christoffel-Minkowski problem solved for half-space.

New central elements found in a quantum algebra related to knot theory.

This paper studies spectral properties of spheres with one equator.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…

Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_Θ)_{Θ>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer normals with speeds $-\log(F/f)$. For quite general functions $F$ of the principal curvatures of …

Let $ψ$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $ψ$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness…

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solutio…

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some act…

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $σ_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $σ_k-$curvature. The $σ_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the…

In this paper we establish stability results for symmetric spaces of noncompact type under Ricci flow, i.e. we will show that any small perturbation of the symmetric metric is flown back to the original metric under an appropriately rescaled Ricci flow. It will be important for us which smallness assumptions we have to…

Inverse function theorem and homotopy description for L-infinity bundles.

New proofs show smallest non-cyclic quotients for braid and mapping class groups.

R. Lutz introduced the notion of $\varGamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\varGamma$ is a finite abelian group. In the present article, as $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$, we give the automorphisms $\tildeσ_3, \tildeτ_3$ of order $3$ o…

We consider the cohomology group $H^1(Γ, ρ)$ of a discrete subgroup $Γ\subset G=SU(n, 1)$ and the symmetric tensor representation $ρ$ on $S^m(\mathbb C^{n+1})$. We give an elementary proof of the Eichler-Shimura isomorphism that harmonic forms $H^1(Γ\backslash G/K, ρ)$ are $(0, 1)$-forms for the automorphic holomorphic…

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describi…

Unified flow solves $L^p$ Christoffel-Minkowski problem for $p>1$.

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new struc…