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Research:


My research deals with PDEs and, more generally, analytical methods related to geometry and topology as well as methods coming from mathematical physics. Particular areas of specialisation include:

Spectral geometry; extremal eigenvalue problems.
I have been working on isoperimetric inequalities and multiplicity bounds for various eigenvalue problems. I am also intersted in extremal eigenvalue problems. One of the past projects is concerned with the development of direct methods for such extremal problems on Riemannian surfaces and understanding of possible singularities of extremal metrics.

Harmonic maps and their generalisations.
Another interest of mine is general equations of the type of harmonic maps, related analytic phenomena, and their applications to geometry. One of my contributions here is concerned with exploring the relationship between bubble convergence and the curvature concentration. In another paper I developed a number of applications of the so-called pseudo-harmonic maps in conformal/complex geometry.

Moduli spaces of solutions to elliptic PDEs.
In my PhD thesis and shortly after I have been studying moduli spaces formed by solutions of PDEs on mappings between manifolds together with Sergei Kuksin. My favourite application of these results is to the topology of the evaluation map; it establishes a relationship between the symplectic version of the Gottlieb vanishing phenomenon and the occurrence of rational curves.



Preprints and publications:
  • (with A. Hassannezhad, I. Polterovich) Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound. Yuri Safarov's memorial volume. Journal of Spectral Theory, to appear. arXiv:1510.07281

  • (with V. Apostolov, D. Jakobson) An extremal eigenvalue problem in Kahler geometry. Paul Gauduchon's anniversary volume. J. Geom. Phys. 91 (2015), 108-116. arXiv:1411.7725

  • (with A. Hassannezhad) Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appear. arXiv:1407.0358

  • On multiplicity bounds for Schrodinger eigenvalues on Riemannian surfaces. Anal. PDE 7 (2014), 1397-1420. arXiv:1310.2207

  • (with M. Karpukhin, I. Polterovich) Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. Ann. Inst. Fourier 64 (2014), 2481-2502. arXiv:1209.4869

  • Sub-Laplacian eigenvalue bounds on CR manifolds. Comm. Partial Differential Equations 38 (2013), 1971-1984. arXiv:1202.5465

  • Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258 (2014), 191-239. arXiv:1103.2448

  • Curvature and bubble convergence of harmonic maps. J. Geom. Anal. 23 (2013), 1058-1077. arXiv:1005.3783

  • On the concentration-compactness phenomenon for the first Schrodinger eigenvalue.
    Calc. Var. Partial Differential Equations 38 (2010), 29-43. arXiv:0904.3264

  • (with N. Nadirashvili) On the first Neumann eigenvalue bounds for conformal metrics.
    Topics around the Research of Vladimir Maz'ya. II, 229-238, "International Mathematical Series" 12, Springer, 2010.

  • (with D.Kotschick) Fibrations and fundamental groups of Kaehler--Weyl manifolds.
    Proc. Amer. Math. Soc. 138 (2010), 997-1010. arXiv:0811.1952

  • On geodesic homotopies of controlled width and conjugacies in isometry groups.
    Groups Geom. Dyn. 7 (2013), 911-929. PDF arXiv:0709.3469

  • On pseudo-harmonic maps in conformal geometry. 2
    Proc. London Math. Soc. 99 (2009), 168-194. Based on the preprint arXiv:0705.3821

  • On the topology of the evaluation map and rational curves.1
    Internat. J. Math. 19 (2008), 369-385. PDF arXiv:math/0603255

  • A note on Morse inequalities for harmonic maps with potential and their applications.
    Ann. Global Anal. Geom. 33 (2008), 101-113. PDF

  • (with S.Kuksin) Quasi-linear elliptic differential equations for mappings of manifolds, II.
    Ann. Global Anal. Geom. 31 (2007), 59-113. PDF

  • Elements of qualitative theory of quasilinear elliptic partial differential equations for mappings valued in compact manifolds. PhD Thesis, Heriot-Watt University, 2004.

  • On the compactness property of the quasilinearly perturbed harmonic map equation.
    Sbornik: Mathematics 194 (2003), no. 7, 1055-1068. PDF

  • (with S.Kuksin) Quasilinear elliptic differential equations for mappings between manifolds, I. (Russian) Algebra i Analiz 15 (2003), no. 4, 1-60; translation in St.Petersburg Math. Journal 15 (2004), 469-505.

  • The compactness of the set of solutions of the quasilinearly perturbed harmonic map equation. Russian Math. Surveys, 57 (2002), 995-996.



Slides of some talks:

Footnotes:
1. The pdf-file on this web-site is a more recent version of the paper than the one on arXiv.org. It contains a remark made by Dieter Kotschick concerning the properties of the evaluation map on the whole 2-skeleton of Symp.
2. The journal version is shorter than the one on arXiv.org.

http://www1.maths.leeds.ac.uk/~pmtgk/research.html
Last modified: 17 Aug 2016