By Steffen Fröhlich

This e-book is meant for complicated scholars and younger researchers attracted to the research of partial differential equations and differential geometry. It discusses straightforward recommendations of floor geometry in higher-dimensional Euclidean areas, specifically the differential equations of Gauss-Weingarten including numerous integrability stipulations and corresponding floor curvatures. It encompasses a bankruptcy on curvature estimates for such surfaces, and, utilizing effects from capability thought and harmonic research, it addresses geometric and analytic ways to determine the lifestyles and regularity of Coulomb frames of their basic bundles, which come up as serious issues for a useful of overall torsion.

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**Additional resources for Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces: Topics from Differential Geometry and Geometric Analysis of Surfaces**

**Example text**

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.