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Igor Prokhorenkov

Publications and source records attributed to Igor Prokhorenkov.

7 recordsLinked to original sources
arXiv2017

Singular Riemannian flows and characteristic numbers

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of $M$ as the sum of residues associated to the infinitesimal foliation at the components of the singular stratum of the flow.

math.DG↗
arXiv2008

Natural Equivariant Dirac Operators

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.

math.DG↗
arXiv2008

Semiclassical Asymptotics on Manifolds with Boundary

We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers.

math.SP↗
arXiv2006

Witten deformation and the equivariant index

Let $M$ be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on $M$. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.

math.DG↗
arXiv2003

Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_{s}=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semi-classical limit $s\to \infty $. We show how to use Witten's method to compute the index of $D$ by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator $Z$. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a spin$^{c}$ manifold to maps between its even and odd spinor bundles. The paper contains a list of the current literature on the subject.

math.DG↗
arXiv1998

Hodge theory and cohomology with compact supports

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_\mu$ associated to a measure $d\mu$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_\mu$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.

math.DG↗