By Yuan-Jen Chiang
Harmonic maps among Riemannian manifolds have been first verified 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 Nineteen Nineties. Yang-Mills fields, the severe issues of Yang-Mills functionals of connections whose curvature tensors are harmonic, have been explored by means of a couple of physicists within the Fifties, and biharmonic maps (generalizing harmonic maps) have been brought via Guoying Jiang in 1986. The e-book 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) that have been first studied by way of 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 subject matters mentioned are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.
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Extra info for Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields
120, 241]). Lawson also showed that a compact embedded minimal surface in S 3 separates it into two diffeomorphic components. In all his examples, these components have equal volumes. But  produced new minimal surfaces in S 3 separating it into two components of different volumes. Smyth  showed that every flat torus T3 contains minimally embedded surfaces of any high genus. (cf. ) Bryant  proved that any compact Riemann surface M can be conformally harmonically immersed in S 4 . D) produce embedded minimal surfaces of genus p in S 3 .
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.
We call such a ball geodesically small. N; h/, then f is smooth and harmonic on M . N; h/ of finite energy extends to a harmonic map on D 2 . M; g/ ! M; N / arbitrarily p q near f0 which is harmonic on M E but not on M. Following Brezis, Coron, and Lieb , consider maps R3 ! S 2 with point singularities. ; ap be p points in R3 and d1 ; ; dp be p non-zero integers P Let a1 ; such that di D 0. f; ai / is the P Brouwer degree of f restricted to a small ai -centered 2-sphere; E ¤ ;; since di D 0.