By Luigi Bianchi

In queste ''Lezioni di geometria differenziale'' i soggetti più elevati dell'aritmetica (forme quadratiche, corpi algebrici e loro ideali, rapporti reciproci fra l. a. teoria dei numeri e quelle delle funzioni analitiche) sono trattati con una rara ampiezza e ricchezza di sviluppi. Si deve all'opera del Bianchi, matematico di fama mondiale, il grande progresso della Geometria differenziale.

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In particular, we have in mind to confront some of the methods and results from this field of geometric analysis with the concepts of extrinsic differential geometry which we developed in the first chapter. 2 Curvature Estimates 37 This plan must be left incomplete due to its complexity. We will therefore concentrate on some “light” versions of curvature estimates and their immediate consequences, and we will only discuss briefly more profound approaches and methods. ;22 /L ;12 of the normal curvature tensor from Sect.

G. Blaschke and Leichtweiß [12] for more details on this famous identity connecting analysis, topology and differential geometry. And the conformally invariant functional ZZ jS jW d ud v B measures the total normal curvature of the surface. In Sakamoto [101] we find the probably first investigations on critical points of this functional, and this should open new fields in classical differential geometry. 3 The Special Case of Holomorphic Minimal Graphs We want to specify the foregoing estimate jS jW Ä 2jHj2 W KW in case of holomorphic minimal graphs.

0 with a fixed axis in space. n C 1/ : d2 sin4 ! In particular, if the surface is defined over the whole R2 ; then it is a plane. -condition with the curvature of the normal bundle of complete minimal graphs is not known to us. -condition. A refinement of Osserman’s proof together with applications of potential theoretic methods enabled us in Bergner and Fr¨ohlich [8] to prove the following curvature estimate for graphs with prescribed H¨older continuous mean curvature vector. 11. n C 1/-dimensional unit sphere.