By Yuan-Jen Chiang

Harmonic maps among Riemannian manifolds have been first tested by way of James Eells and Joseph H. Sampson in 1964. Wave maps are harmonic maps on Minkowski areas and feature been studied because the Nineties. Yang-Mills fields, the serious issues of Yang-Mills functionals of connections whose curvature tensors are harmonic, have been explored by means of a couple of physicists within the Nineteen Fifties, and biharmonic maps (generalizing harmonic maps) have been brought by means of Guoying Jiang in 1986. The ebook provides an summary of the $64000 advancements made in those fields for the reason that they first got here up. additionally, it introduces biwave maps (generalizing wave maps) that have been first studied through the writer in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated through Toshiyuki Ichiyama, Jun-Ichi Inoguchi and Hajime Urakawa in 2008. different issues mentioned are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.

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**Example text**

For instance, a map f W S 3 ! S 3 ; S 2 /; but can not be L21 -approximated by C i maps. N / ¤ 0 and i Ä p < i C 1 Ä m; where B m is the closed Euclidean unit m-ball. M; N /I the latter is non-separable for m 2. In terms of partial derivatives in theR charts, the energy of an L21 -map is well defined. M; N / ! M; Rk /. M; N / ! M; g/ ! N; h/. f / Ä E. M; N / for which f D on M nU . x//, where is the orthogonal projection of Rk onto N (well-defined for sufficiently small t). M; N /. gij /dx is the volume form of M , f ˛ ; ˛ are the components of f; in Rk ; and A is the second fundamental form of the embedding of N in Rk (cf.

If every compact subset of U has a neighborhood U 0 on which there is a function k W U 0 ! R with positive definite Hessian Ddk, we say that U N is convex supporting. M; g/ to a convex supporting domain U is constant. M / of the harmonic map f , then k ı f is subharmonic and thus constant. M /. N; h/ is convex supporting, then any harmonic map W S i 1 ! N; h/ is constant for i 3. In fact, such a map lifts as a harmonic map Q W S i 1 ! NQ ; h/. M; g/ ! N; h/ is smooth. N; h/ whose universal covers are convex supporting.

Furthermore, f W U ! N is a smooth harmonic map. R/ ! N be a sequence of critical maps of E˛ for ˛ ! R/ is a disk with radius R and center the origin. f˛ / < ; then f˛ ! R=2/ ! N is a smooth harmonic map. 1). Let f˛ W M ! 1 C K 2 /˛ as in (L1). Letting M D U in (L2), we can select a subsequence ˇ ! M fx1 ; ; xl g; N / with f W M ! N is harmonic. We want to show that fˇ ! M; N /. 20 1 Harmonic Maps Let D. / be a small disk centered at xi in M with radius , where is sufficiently small such that xj 2 D.