Some Problems On Finsler Geometry And L～2 Harmonic Forms

The content of this paper is divided into two parts. In the first part, which contains the first and the second chapters, we investgate some problems of Finsler Geometry. In the second part, which contains the third chapter, we study the existence of L~2 harmonic forms.In chapter one, we obtain a generalization of Schwarz-Ahlfors lemma in Finsler Geometry and generalize the result in [GIP].The classical lemma of Schwarz states:" Let Φ: D → D be a holomorphic mapping of the unit disc to the unit disc such that Φ(0) = 0.Then |Φ(z)| ≤ |z|, and Φ'(0) = 0 . "Introducing the Poincare - Bergman metric ds~2 = (dzdz|-)/((1-|z|~2))~2 for D Schwarz's lemmabecomes equivalent to:"If Φ:D→D is holomorphic then Φ~*ds~2≤ ds~2."The Poincare — Bergman metric makes D a Kahler manifold of constant section curvature-4.With this remark,Ahlfors gave the following extension of Schwarz's lemma:"Let M be a one-dimensional Kahler manifold with metric ds~2_M whose Gauss curvatureis bounded above by a negative constant —B.Let D_a be an open disc of radius a with metricgiven by ds_D~2 =(4a~2dzdz|-)/A(a-|z|~2)~2' (which makes D_a a Kahler manifold of constant curvature-A < 0). If Φ: D_a → M is holomorphic then Φ~*ds_M~2 ≤-A(ds_D~2)/B."If A < B the mapping is distance decreasing. In the general case we will say that the mapping is distance decreasing up to a constant.