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V. M. Gichev

Publications and source records attributed to V. M. Gichev.

10 recordsLinked to original sources
arXiv2012

Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

Let $M=G/H$ be a compact connected isotropy irreducible Riemannian homogeneous manifold, where $G$ is a compact Lie group (may be, disconnected) acting on $M$ by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori $\bbR^n/\bbZ^n$ with the natural action on itself extended by the finite group generated by all transpositions of coordinates and inversions in circle factors. We say that $u$ is a polynomial on $M$ if it belongs to some $G$-invariant finite dimensional subspace $\cE$ of $L^2(M)$. We compute or estimate from above the averages over the unit sphere $\cS$ in $\cE$ for some metric quantities such as Hausdorff measures of level set and norms in $L^p(M)$, $1\leq p\leq\infty$, where $M$ is equipped with the invariant probability measure. For example, the averages over $\cS$ of $\|u\|_{L^p(M)}$, $p\geq2$, are less than $\sqrt{\frac{p+1}{e}}$ independently of $M$ and $\cE$.

math.CA↗
arXiv2009

Invariant function algebras on compact commutative homogeneous spaces

Let $M$ be a commutative homogeneous space of a compact Lie group $G$ and $A$ be a closed $G$-invariant subalgebra of the Banach algebra $C(M)$. A function algebra is called antisymmetric if it does not contain nonconstant real functions. By the main result of this paper, $A$ is antisymmetric if and only if the invariant probability measure on $M$ is multiplicative on $A$. This implies, for example, the following theorem: if $G^{\mathbb C}$ acts transitively on a Stein manifold $\cal M$, $v\in{\cal M}$, and the compact orbit $M=Gv$ is a commutative homogeneous space, then $M$ is a real form of $\cal M$.

math.FA↗
arXiv2008

Some remarks on spherical harmonics

The article contains several observations on spherical harmonics and their nodal sets: a construction for harmonics with prescribed zeroes; a kind of canonical representation of this type for harmonics on $\bbS^2$; upper and lower bounds for nodal length and inner radius (the upper bounds are sharp); precise upper bound for the number of common zeroes of two spherical harmonics on $\bbS^2$; the mean Hausdorff measure on the intersection of $k$ nodal sets for harmonics of different degrees on $\bbS^m$, where $k\leq m$ (in particular, the mean number of common zeroes of $m$ harmonics).

math.CA↗
arXiv2007

Maximal ideal spaces of invariant function algebras on compact groups

The paper contains a description of the maximal ideal spaces (spectra) $\cM_A$ of bi-invariant function algebras $A$ on a compact group $G$. There are natural compatible structures in $\cM_A$: it is a compact topological semigroup with involution, polar decomposition, and analytic structure. The paper contains a description of $\cM_A$ and related results on function algebras; for example, a bi-invariant function algebra on a connected compact Lie group is antisymmetric if and only if the Haar measure of its maximal torus is multiplicative on it. Some results are extended to the case of compact commutative homogeneous spaces. As a consequence, we get an infinite dimensional version of the Hilbert--Mumford criterion for commutative (as homogeneous spaces) orbits of compact connected Lie groups, where one parameter semigroup is replaced by a finite sequence of them. Keywords: Maximal ideal space, invariant function algebra, complex Lie semigroup, commutative homogeneous space, Hilbert--Mumford criterion

math.FA↗
arXiv2007

Orbits of tori extended by finite groups and their polynomial hulls: the case of connected complex orbits

Let $V$ be a complex linear space, $G\subset\GL(V)$ be a compact group. We consider the problem of description of polynomial hulls $\wh{Gv}$ for orbits $Gv$, $v\in V$, assuming that the identity component of $G$ is a torus $T$. The paper contains a universal construction for orbits which satisfy the inclusion $Gv\subset T^\bbC v$ and a characterization of pairs $(G,V)$ such that it is true for a generic $v\in V$. The hull of a finite union of $T$-orbits in $T^\bbC v$ can be distinguished in $\clos T^\bbC v$ by a finite collection of inequalities of the type $\abs{z_1}^{s_1}...\abs{z_n}^{s_n}\leq c$. In particular, this is true for $Gv$. If powers in the monomials are independent of $v$, $Gv\subset T^\bbC v$ for a generic $v$, and either the center of $G$ is finite or $T^\bbC$ has an open orbit, then the space $V$ and the group $G$ are products of standard ones; the latter means that $G=S_nT$, where $S_n$ is the group of all permutations of coordinates and $T$ is either $\bbT^n$ or $\SU(n)\cap\bbT^n$, where $\bbT^n$ is the torus of all diagonal matrices in $\rU(n)$. The paper also contains a description of polynomial hulls for orbits of isotropy groups of bounded symmetric domains. This result is already known, but we formulate it in a different form and supply with a shorter proof.

math.CV↗
arXiv2007

On Geometry of Flat Complete Strictly Causal Lorentzian Manifolds

A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the cone of positive definite matrices. Two manifolds are equivalent if and only if their signatures coincides and the corresponding parabolas are equal (up to a suitable automorphism of the cone and an affine change of variable). Also, we give necessary and sufficient conditions, which distinguish parabolas of this type among all parabolas in the cone.

math.MG↗
arXiv2005

On complete affine structures in Lie groups

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the multiplication by left defines a representation of the underlying Lie algebra). An affine structure (and the corresponding left symmetric algebra) is complete if $G$ is affinely equivalent to $\mathfrak g$. By the main result of this paper, a complete left symmetric algebra admits a canonical decomposition: there is a Cartan subalgebra $\mathfrak h$ such that the root subspaces for the representations $L$ (by left multiplications) and $\ad$ coincide. Then operators $L(x)$ and $\ad(x)$ have equal semisimple parts for all $x\in\mathfrak h$. This decomposition is unique. For simple complete left-symmetric algebras whose canonical decomposition consists of one dimensional spaces we define two types of graphs and prove some their properties. This makes possible to describe, for dimensions less or equal to 5, these graphs and algebras.

math.DG↗
arXiv2005

A note on common zeroes of Laplace--Beltrami eigenfunctions

Let $\De u+\la u=\De v+\la v=0$, where $\De$ is the Laplace--Beltrami operator on a compact connected smooth manifold $M$ and $\la>0$. If $H^1(M)=0$ then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$, $H^1(M)\neq0$ implies the existence of a pair $u,v$ as above that has no common zero.

math.MG↗
arXiv2005

On flat complete causal Lorentzian manifolds

We describe up to finite coverings causal flat affine complete Lorentzian manifolds such that the past and the future of any point are closed near this point. We say that these manifolds are strictly causal. In particular, we prove that their fundamental groups are virtually abelian. In dimension 4, there is only one, up to a scaling factor, strictly causal manifold which is not globally hyperbolic. For a generic point of this manifold, either the past or the future is not closed and contains a lightlike straight line.

math.MG↗