By Yuan-Jen Chiang

Harmonic maps among Riemannian manifolds have been first validated through 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 severe issues of Yang-Mills functionals of connections whose curvature tensors are harmonic, have been explored by way of a number of physicists within the Nineteen Fifties, and biharmonic maps (generalizing harmonic maps) have been brought via Guoying Jiang in 1986. The publication offers an outline of the real advancements made in those fields for the reason that they first got here up. in addition, it introduces biwave maps (generalizing wave maps) which have been first studied via the writer in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated by means of 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**

B) For m 2 an L21 -map may not be continuous, and there is no natural concept of its homotopy class. M; g/ ! M / ! M; N /. That was shown by Schoen and Yau [323] for m D 2. A detailed proof was given by Burstall [47] and White [402–404] for m 2. The main idea is that even though f does not restrict to a continuous map on a given loop ˛ in M , it does so restrict on almost every loop in a tubular neighborhood of ˛; their images are homotopic and thereby define f Œ˛. N / D 0 for i 2. Then the homotopy classes of maps M !

N /. If m 3, f is not necessarily continuous and the size of its singular set can be estimated as follows. Schoen and Uhlenbeck [320–322] obtained the main partial regularity theorem for harmonic maps in the general case. M / is contained in a single chart of N . From now on, we focus on the work of Schoen and Uhlenbeck and try to outline their main results. N n ; h/ be Riemannian manifolds of dimension m and n. By the Nash imbedding theorem, we can assume that N Rk is isometrically embedded in the Euclidean space.

M; g; J / ! N; h; J / be a smooth map between almost hermitian manifolds. M / ! M / ! M / ! M / ! M / ! M / D @f N C @N fN: We get @N fN D @f ; @fN D @f N D A map f is holomorphic (resp. , @f 0 (resp. , @f D 0/. A map f is ˙ holomorphic if it is holomorphic or antiholomorphic. f / be their integrals. f / and f is holomorphic (resp. f / D 0 (resp. f / D 0). The relationship between ˙ holomorphic maps and harmonic maps can describe as follows. N; h; J / be almost hermitian manifolds. , the codifferential of !