Download Geometry of Harmonic Maps by Yuanlong Xin (auth.) PDF

By Yuanlong Xin (auth.)

Harmonic maps are options to a typical geometrical variational prob­ lem. This thought grew out of crucial notions in differential geometry, corresponding to geodesics, minimum surfaces and harmonic services. Harmonic maps also are heavily with regards to holomorphic maps in numerous advanced variables, to the idea of stochastic techniques, to nonlinear box thought in theoretical physics, and to the speculation of liquid crystals in fabrics technological know-how. in the past thirty years this topic has been built largely. The monograph is not at all meant to offer an entire description of the speculation of harmonic maps. for instance, the publication excludes a wide a part of the idea of harmonic maps from 2-dimensional domain names, the place the equipment are particularly diversified from these mentioned the following. the 1st bankruptcy comprises introductory fabric. a number of an identical definitions of harmonic maps are defined, and engaging examples are offered. a variety of very important homes and formulation are derived. between them are Bochner-type formulation for the strength density and the second one varia­ tional formulation. This bankruptcy serves not just as a foundation for the later chapters, but in addition as a short creation to the idea. bankruptcy 2 is dedicated to the conservation legislations of harmonic maps. Em­ phasis is put on functions of conservation legislation to the mono tonicity formulation and Liouville-type theorems.

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It might be of interest to study harmonic p-forms with values in a vector bundle. Let IT : E --+ M be a Riemannian vector bundle over a Riemannian manifold, and w a p-form with values in E, which may be viewed as a cross-section of the vector bundle APT M (9 E. Similarly, we can introduce its stress-energy tensor. 17) w 0 w(X, Y) = (ixw, iyw) , where X, Y E f(T M), and ixw denotes the inner product of p-form w and vector field X. This is a (p - I)-form and is defined by ixw(X I ,'" ,Xp-d d~r w(X,X I ,··· ,Xp- l ), where Xl,'" ,Xp- l E f(T M).

The Bergman metric corresponds to the canonical metric in G~ m(C). The second and third types of bounded symmetric domains ~II(n); ~III(n) are totally geodesic submanifolds of G~ m(C). The fourth type of bounded symmetric domains ~IV( n) are ide~tified with 2 (see [Hua3] and [Wo]). Therefore, by the moving frame method we can c~mpute the curvature tensors and estimate the bounds of the sectional curvatures and of the Ricci curvatures. : 2) Dim. 2mn n(n + 1) n(n - 1) 2n Sec. Curvature -4:::; K:::; 0 -4:::; K:::; 0 -2:::; K:::; 0 -2:::; K:::; 0 Ric.

For any v E rU-1T N), there exists a family of maps ft, -c < t < c, satisfying it may be viewed as a map from (-c,c) x Minto N. 13) we have EUt)\ = dd: t t=o J[M / v, V \ ~a. rUt)) It=O * 1. 3) Choose an orthonormal frame field {ed with Ve;eilp = 0 near p E M. Then V ~ BBt = V ~ei = Ve BBt = O. We have at the point p 8~ 8t o I V~ rUt) at = V ~(Veodft)ei at I = (V ~ Veodft)ei a. 4) U*ei,v)f*ei + RNU*ei,v)f*ei. 5) This formula is due to R. T. Smith and E. Mazet [Sm1] [Ma]. 7) Noting the properties of the trace-Laplacian operator and the curvature tensor, we know that I(v, w) is a bilinear form defined on r(f-lTN), and J j is a strongly elliptic differential operator of second order on r(f-lTN).

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