Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Curves on normal surfaces

We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces.

     

Curves on surfaces and surgeries

In this paper, we introduce a new surgery operation on the set of collections of curves, on a closed oriented surface, whose complement is a topological disk. We prove that any two such collections can be connected by a sequence of surgeries.     

On the existence of nontrivial extremal metrics on complete noncompact surfaces

In this paper, we consider the 2-dimensional local Calabi flow on a complete noncompact surface . Then, based on the Harnack-type estimate, we show the long-time existence and asymptotic convergence of a subsequence of solutions of such a flow on with and bounded from above by a negative constant on a ball. For its applications, this will lead to the existence of extremal metrics on a complete noncompact surface of finite topological type. In particular, there exists an extremal metric of nonconstant Gaussian curvature on or Received: 21 June 2001 / 18 January 2002 / Published online: 27 June 2002 Research supported in part by NSC and NCTS.     

Parabolic stable Higgs bundles over complete noncompact Riemann surfaces

LetM be an open Riemann surface with a finite set of punctures, a complete Poincaré-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle, and the existence of a Hermitian-Einstein metric on the bundle is established. Project surpported partially by the National Natural Science Foundation of China (Grant No. 19701034).     

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on ?-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical ?-periodic surface with a boundary are recurrent.     

Fixed-point theories on noncompact manifolds

This paper gives three noncompact variants of Lefschetz–Nielsen fixed-point theory parallel to developments that have occurred in surgery theory. Thus, we study when a proper map can be properly homotoped, or boundedly homotoped, or even homotoped in a C^r bounded fashion to a fixed-point free map. As an example, the universal cover of a compact aspherical manifold always has a fixed-point free self-diffeomorphism C^r close to the identity for all r (although this is not the case, in general, for arbitrary infinite covers of such manifolds, or general universal covers).     

Infinitesimal transformations on noncompact manifolds

Summary The object of study in this paper is the A_x-operator associated to a Killing field (resp. to an infinitesimal analytic transformation) on Riemannian (resp. Kähler) complete but noncompact manifolds. We recall (§ 1) Kostant's and Lichnéroivicz's Theorems for that operator in the compact case and we dedicate, § 2 and § 3 to study all the simply-connected noncompact manifolds in which similar results to that theorems do not hold, we give examples of all these cases. § 4 is dedicated to give sufficient conditions in order to ensure the validity of similar results to that above mentioned, on complete noncompact manifolds. The hypothesis used refer to vector fields whose norm are either pointwise bounded or globally bounded.Partially supported by C.A.I.C.Y.T. 1085-84.     

Diffeomorphism groups on noncompact manifolds

A quite general theory of manifolds of maps between open manifolds of bounded geometry is constructed. The results of the paper can be applied to the case of compact manifolds, but the method used differs from the traditional approach of Ebin and Marsden. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 41–65.