Removal of Isolated Singularities of Generalized Quasiisometries on Riemannian Manifolds
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REMOVAL OF ISOLATED SINGULARITIES OF GENERALIZED QUASIISOMETRIES ON RIEMANNIAN MANIFOLDS UDC 517.5
D. P. Ilyutko and E. A. Sevostyanov
Abstract. For mappings with unbounded characteristics, we prove theorems on removal of isolated singularities on Riemannian manifolds. We prove that if a mapping satisfies certain prototype inequality of absolute values and its quasiconformity characteristic has a majorant of finite average oscillation at a fixed singular point, then it has a limit at that point.
CONTENTS 1. Introduction . . . . . . 2. Auxiliary Lemmas . . 3. Proofs of Main Results References . . . . . . .
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611 614 618 620
Introduction
According to the well-known complex-analysis theorem on the removal of isolated singularities, any bounded analytic function ϕ : D \ {z0 } → C of a domain D \ {z0 } ⊂ C has a limit at the point z0 (see [19, Th. 1 , § 6, Sec. 24, Chap. II]). In n-dimensional spaces, that result is generalized for maps with bounded distortions (see [8, Corollary 4.5] and [13, Th. 2.9, Chap. III], see [11, 12] as well). Further generalizations are obtained later. In particular, in [17, 18], analogs of the Sokhotskii–Weierstrass theorem are obtained for socalled ring Q-maps, i.e., maps such that their definition is based on the control distortion property of the module of curve families. Here, the function Q is responsible for the “damage” of the module; in general, this function is assumed to be unbounded. Usually, results for such maps are substantial if the growth of the function Q in a neighborhood of the fixed isolated point is weak: e.g., this is true for logarithmic-type singularities, functions with finite mean oscillations, etc. (see ibid.). Note that analytic functions correspond to the case where Q ≡ 1, while bounded-distortion maps correspond to the case where Q ≡ const. The majority of map classes known nowadays are classes of ring Q-maps as well under reasonable (not very strong) restrictions for the characteristic of the quasiconformality, smoothness of maps, and measure of the set of their branch points (see [16]). In this paper, we show that an analog of the theorem on the removal of isolated singularities for ring Q-maps holds not only in the Euclidean n-dimensional space, but on Riemannian manifolds as well, and the required restrictions for the function Q are almost the same as in the Euclidean case. Also, restrictions for the manifolds themselves are added; they are specified below. Pass to definitions and main results. Notions introduced below can be found, e.g., in [7, 19]. Recall that the n-dimensional topological manifold Mn is the Hausdorff topological space with a countable base such that each of its points has a neighborhood homeomorphic
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