Systems are classified of partial differential equations whose solution classes are stable in the uniform norm. Properties of solutions are studied for multidimensional analogs of the Beltrami equation describing quasiregular functions of a complex variable. The theory is developed of quasiregular mappings in several spatial variables and a stability theorem is proved for this class. Mappings with bounded distortion on the Heisenberg groups with the Carnot–Caratheodory metric, as well as on more general two-step niltpotent Lie groups, have been defined and studied. Together with C. Croke and V. A. Sharafutdinov, a local boundary rigidity theorem is proved for Riemannian manifolds with above bounded curvature. Together with V. A. Sharafutdinov, a finiteness theorem is proved for infinitesimal isospectral deformations of manifolds whose geodesic flow is of Anosov type, and uniqueness is proved for solutions to some integral geometry problems on these manifolds.
Main publications:
Croke C., Dairbekov N. S., Sharafutdinov V. A. Local boundary rigidity of a compact Riemannian manifold with curvature bounded above // Trans. Amer. Math. Soc. 2000, 352, 3937–3956.
