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By Jürgen Jost

In this e-book, I current an elevated model of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Düsseldorf, June, 1986. The name "Nonlinear tools in complicated geometry" already shows a mix of innovations from nonlinear partial differential equations and geometric innovations. In older geometric investigations, frequently the neighborhood elements attracted extra cognizance than the worldwide ones as differential geometry in its foundations offers approximations of neighborhood phenomena via infinitesimal or differential buildings. the following, all equations are linear. If one desires to contemplate international features, although, frequently the presence of curvature Ieads to a nonlinearity within the equations. the easiest case is the only of geodesics that are defined through a approach of moment ordernonlinear ODE; their linearizations are the Jacobi fields. extra lately, nonlinear PDE performed a progressively more pro~inent röle in geometry. allow us to Iist essentially the most very important ones: - harmonic maps among Riemannian and Kählerian manifolds - minimum surfaces in Riemannian manifolds - Monge-Ampere equations on Kähler manifolds - Yang-Mills equations in vector bundles over manifolds. whereas the answer of those equations frequently is nontrivial, it may well Iead to very signifi­ cant ends up in geometry, as ideas offer maps, submanifolds, metrics, or connections that are unusual by way of geometric homes in a given context. a majority of these equations are elliptic, yet frequently parabolic equations are used as an auxiliary software to unravel the elliptic ones.

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Extra info for Nonlinear Methods in Riemannian and Kählerian Geometry: Delivered at the German Mathematical Society Seminar in Düsseldorf in June, 1986

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2). We call D the metric complex connection for E. ;)i,i=l, ...... 4) In particular, it is of type (1,1). Conversely, given a complex vector bundle E over a complex manifold M, the existence of a holomorphic structure on E is equivalent to the existence of a 8-operator ä : 0° (E) satisfying the product rule (f E -+ 0°· 1(E) coo (M,a'), u E 0°(E)) ä(fu) = ät · u + f äu, with (extended as ä : 0°· 1(E) -+ 0°· 2 (E)). If E carries a Hermitian metric, this in turn is equivalent to the existence of an U(r)-connection with curvature of type (1,1).

30) = 0 <===> d* F = 0 = dF, and the existence and uniqueness of a solution follows from Hodge's Theorem, cf. 1. Still, the connection Ais not unique, since only F, but not A remains invariant under the action of g. l is a trivial bundle); so that with s = e", with da= 0. l 1 (Ad E) 0 s*(A) = A+ a, (Ad E) with a =du (remember that 41 and consequently 9 acts transitively on the space of Yang-Mills connections. Actually, in the present case, the only possible Abelian G is U(l) = 80(2) (neglecting the trivial group SO(l)).

Ft) ßtP(F't) = kP(DtTJ,Ft, ... ,Ft) = d(kP(TJ, Ft, ... , Ft) since Dt Ft = 0 by Bianchi's identity Consequently is cohomologous to zero. qed. The lemma establishes the Weil homomorphism w : Algebra of graded invariant polynomials ---+ H 2 * (M) w(P) = [P(F)] The Chern classes of E, then, are defined as where pi is the ith elementary polynomial. ,- are the eigenvalues of ~: 1 F 1 l. g. ;, + ... ;,)x); ... n in particular c1(A"'T'M) = c1(M) (m = dima: M). 1 is a 2-form. ~x) 59 1. g. in [W]. Suppose M is a differentiable manifold which has a complex as well as a Hermitian structure.

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