Igor Prokhorenkov

Publications and Preprints

The most recent items are listed first.

• (with Ken Richardson) Singular Riemannian flows and characteristic numbers, accepted in Ann. Glob. Anal. Geom. Click here to get a version from the arXiv.
  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. .
  
  
• (with Ken Richardson) Perturbations of basic Dirac operators on Riemannian foliations, International Journal of Mathematics, volume 24 (2013) Click here to get a version from the arXiv.
  Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.
  
  
• (with Ken Richardson) Natural Equivariant Dirac Operators, Geometriae Dedicata, volume 151 (2011), 411-429. Click here to get a version from the arXiv.
  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.
  
  
• (with Mikhail Shubin) and Nilufer Koldan Semiclassical Asymptotics on Manifolds with Boundary, Contemporary Mathematics, volume 483 (2009), 239-266. Click here to get a version from the arXiv.
  We have found 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 have modified and used 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)
  
  
• (with Ken Richardson) Witten deformation and the equivariant index Ann. Glob. Anal. Geom., volume 34 (2008), 301-238 . Click here to get a version from the arXiv.
  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.
  
  
• (with Ken Richardson) Perturbations of Dirac operators, J. Geom. Phys. 57 (2006), no. 1, pp. 297-321. Click here to get a version from the arXiv.
  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.
  
  
• (with Bruce Miller) and Peter Klinko Rotation-induced phase transition in a spherical gravitating system, Physical Review E63 (2001), 066131, 1-10. One of the important open questions in astrophysics is to understand the time evolution of globular clusters -- compact spherical groups of as many as hundreds of thousands or millions of stars. Because of the large population of a typical cluster, it is believed that the methods of statistical physics ( the maximum entropy principle) can be used to explain observational data and results of computer simulations. However, the infinite range and a ``zero-distance'' singularity of the gravitational force lead to formidable difficulties in generalizing the results of classical mathematical statistical physics (only applicable to short-range forces). For example, one has to deal with an infinite mass or with the possibility of the gravothermal catastrophe, when all the mass collapses to one point and the entropy becomes unbounded. This paper investigates the influence of rotation on a purely spherical gravitational system. We assume that each particle (star) in the system has specific angular momentum of the same magnitude $l.$ This is a simplified model of a real cluster where only the square of the total angular momentum of all stars is conserved. We undertake a study of the system in the mean field limit, i.e. when the number of particles approaches infinity preserving the total mass, energy, and angular momentum of the system. In order to apply the maximum entropy principle we seek the extreme values of the entropy functional \begin{equation*} S\left[ f\right] =-\iint\limits_{\text{phase space}}\left( f\cdot \ln f\right) drdv \end{equation*} subject to the total mass and mean total energy constraints \begin{equation*} \iint\limits_{\text{phase space}}f(r,v)drdv=1,\text{ }\iint\limits_{\text{ phase space}}f\cdot \left( \frac{1}{2}v^{2}+\frac{l^{2}}{2r^{2}}+\Phi (r)\right) drdv=E, \end{equation*} Here $\Phi (r)$ is the gravitational potential which depends on the mass-velocity density $f(r,v).$ The main mathematical result of the paper is a proof of the existence of a lower bound for the potential energy of the system in terms of the angular momentum $l.$ We apply this bound to prove an upper bound for the entropy of the system in terms of total energy $E$ and $l$ . We have an example showing that our estimates are the best possible. Thus we prove that no gravothermal catastrophe could happen in our model. It would be of considerable interest to rigorously establish the existence of states of (relative) maximum entropy in the mean-field limit. Such states appear to exist in numerical simulations, yet the mathematical proof of their existence requires delicate tools of variational analysis and non-linear PDE's [work in progress].
  Related work. The state of arbitrarily large entropy (``gravothermal catastrophe'') was constructed by Antonov in 1962. Since then only a few mathematically rigorous studies of simplified situations, assuming the existence of some cut-offs of the gravity at the origin and at infinity, exist in literature. Among the most important papers are those of W. Braun and K. Hepp [Commun. Math. Phys. \textbf{56} (1977)], M. K. Kiessling [J. Stat. Physics, \textbf{55 }(1989), 203-257 ] and T. Patmanabhan [Astrophysical Journal suppl. ser.\textbf{\ 71} (1989), p.651-664]. Numerical simulations testing theoretical models were performed by many researchers. Our approach is most closely related to the work of B. Miller and collaborators.
  
  
• (with Edward Bueler) Hodge theory and cohomology with compact support, Soochow J. Math. 28 (2002), no. 1, pp. 33-55. Click here to get a version from the arXiv.
  One of the most important tools of Global Analysis is Hodge theory. For a compact manifold $M$ it provides an isomorphism between $H_{deR}^{p}(M)$, the de Rham cohomology space in dimension $p,$ and the kernel of the Laplacian on smooth $p$-forms on $M.$ The existence of this isomorphism depends on the availability of the spectral gap separating zero eigenvalues of the Laplacian on $M$ from the rest of the spectrum. Such a gap always exists on compact manifolds, where the spectrum consists of discrete eigenvalues. If $M$ is not compact, then the spectral gap often fails to exist. Our paper constructs a Hodge theory for noncompact topologically tame manifolds. A manifold $M$ without boundary is called topologically tame if there exists a compact manifold $N$ with boundary such that $M$ is diffeomorphic to the interior of $N.$ Examples include vector bundles over compact manifolds and manifolds with cylindrical ends. In the paper we study the Hodge theory for the Witten-Bismut Laplacian $\Delta _{\mu }=d\delta _{\mu }+\delta _{\mu }d$ associated to a measure $d\mu =e^{2h}dx,$ where $% h\in C^{\infty }(M)$ and $dx$ is the Riemann-Lebesgue measure.We require that $d\mu $ has sufficiently rapid growth at infinity on $M$ (for instance Gaussian) to ensure the existence of the spectral gap for $\Delta _{\mu }.$ We prove the existence of an isomorphism between the de~Rham cohomology with compact supports of $M$ and the kernel of $\Delta _{\mu }.$ This follows from the construction of a space of forms associated to $\Delta _{\mu }$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing Gaussian growth measures.
  Related work.The results of the paper are motivated by and could serve as extensions of ideas of work of M. F. Atiyah, V. K. Patodi, I. M. Singer [Math. Proc. Camb. Phil. Soc., \textbf{77} (1975), 43-69] and R. Melrose and coworkers on the Atiyah-Patodi-Singer theorem on non-compact manifolds with cylindrical ends. E. Bueler [Trans. Amer. Math. Soc. \textbf{351} (1999), 683-713] examined the case of a complete non-compact manifold $M$ with a Ricci curvature bounded below. He conjectured that the kernel of the Witten Laplacian associated to the ``heat kernel'' measure $d\mu ^{K}=K(t,x_{0},x)dx$ is isomorphic to the de Rham cohomology of $M.$ We hope that the methods of our paper will apply to $d\mu ^{1/K}=\left( K(t,x_{0},x)\right) ^{-1}dx$ in which case the conjecture would hold true. A different approach to Hodge theory on non-compact manifolds is developed in the recent work of Z. M. Ahmed and D. W. Strook [J. Differential Geom. \textbf{54 }(2000), 177-225]. However their results~contrast with the results of our paper. They prove that if $d\mu $ is a measure with such strong decay that the semigroup $e^{t\Delta _{\mu }}$ takes $L_{\mu }^{2}$ into $L^{\infty }$ (``ultracontractivity'') then $\ker \Delta _{\mu }\cong H_{\text{de~R}}(M)$. Among other things, this contraction property implies that the eigenforms of $\Delta _{\mu }$ are actually bounded. Since this is not true for $d\mu =e^{-|x|^{2}}\,dx$ on Euclidean space $\Bbb{R}^{n}$, the Ahmed-Stroock result will not apply to the Gaussian growth/decay case studied by us, for instance. Their result applies to $d\mu =e^{-|x|^{2+\epsilon }}dx$ ($\varepsilon >0$) for $\Bbb{R}^{n}$ and some other noncompact manifolds. Very recent interesting extensions of our results and the results of Ahmed-Strook were published as preprints by F-Z Gong and F-Y Wang.
  
  
• Morse--Bott functions and the Witten Laplacian, Communications in Analysis and Geometry 7(1999), no. 4, pp. 841--918. Twenty years ago E. Witten [J. Differential Geom. \textbf{17 }(1982), 661-692.] published a beautiful new approach to proving Morse inequalities based on the deformation of the de Rham complex. The technique of Witten deformation turned out to be very fruitful in linking a number of topological invariants of smooth manifolds , such as the Euler characteristic, Betti numbers, topological index, Franz-Reidemeister torsion and Novikov numbers to their related analytic realizations in terms of the spectrum of the Laplacian on the critical set of the deformation. Currently, there are over 80 publications on various applications of Witten's method to geometry, topology, and mathematical physics. One of the most general situations where the Witten method is applicable involves studying a smooth compact manifold $N$ without boundary equipped with a Morse-Bott function $h:N\rightarrow \Bbb{R}.$ The critical points of $% h$ form a (disconnected) critical manifold $M$ $\subset N$ and the Hessian $% D^{2}h$ of $h$ is a non-degenerate quadratic form on the normal bundle to $M$ in $N$. In the paper I consider the one-parameter family of deformations of the differential $d$ in the de Rham complex of $M$: \begin{equation*} d\left( \alpha \right) :\omega \mapsto e^{-\alpha h}de^{\alpha h}\omega ,% \text{ }\alpha \in \Bbb{R}_{\geq 0}. \end{equation*} I study the asymptotics as $\alpha \to \infty $ of the discrete spectrum of the \textit{Witten Laplacian} $L(\alpha )=d(\alpha )d^{*}(\alpha )+d^{*}(\alpha )d(\alpha )$ acting on smooth differential forms on $N.$ For a special choice of a Riemannian metric, I prove that asymptotically (as $% \alpha \rightarrow \infty $) the spectrum of the Witten Laplacian separates into the eigenvalues approaching exactly the eigenvalues of the (twisted) Laplacian on a critical manifold $M$, and the large eigenvalues growing faster than $C\cdot \alpha $. The paper also provides estimates on the rate of convergence of the bounded eigenvalues of $L(\alpha )$ to the eigenvalues of the Laplacian on $M$. The main idea of the proof is to generalize the powerful adiabatic limit techniques of R. Mazzeo, R. Melrose [ J. Differential Geom. \textbf{31}(1990), 185-213] and R. Forman [Comm. Math. Phys. \textbf{168 }(1995), 57-116] in order to analyze the spectrum and the eigenspaces of the Witten Laplacian on the tubular neighborhood of the critical manifold $M$. The analysis in the paper is more difficult since the underlying neighborhood is not compact. I use the technical results of the paper to obtain a new Hodge-theoretic proof of the Thom isomorphism and a new analytic proof of the Morse-Bott inequalities. In my proof the difficult probabilistic technique of J.-M. Bismut [J. Differential Geom. \textbf{23 }(1986), 207-240] is replaced by more natural spectral arguments. In addition, one can combine the results of the paper and the independence of the analytic torsion of D. B. Ray and I. M. Singer on the Riemannian metric and on the parameter of the Witten deformation by a Morse-Bott function $h$ to prove a new localization formula for analytic torsion to the critical manifold of $h$ [work in progress].
  Related work.The adiabatic limit technique was first introduced by Witten in 1985. Mazzeo-Melrose (1990) and Forman (1995) applied this technique to fibrations and more general distributions with compact total space. In my paper the adiabatic limit technique was for the first time combined with the Witten method and extended to vector bundles. In 2000 \'{A}lvarez L\'{o}pez and Kordykov generalized the results of Mazzeo-Melrose and Forman to general Riemannian foliations. In 1997 M. Braverman and M. Farber used the Witten method to obtain Novikov inequalities. Their proof only needs the analysis of the kernel of $L(\alpha )$. In 2003 Belfi, Park, and Richardson used Witten deformation to prove an analog of the Hopf theorem in the case of Riemannian foliations. The asymptotic behavior of the Witten deformation of the analytic torsion is actively studied by Braverman, Burghelea, Friedlander, Kappeler, and McDonald.
  
  

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