By Chaohao Gu, Hesheng Hu, Ta-Tsien Li

The overseas convention on smooth arithmetic and the overseas Symposium on Differential Geometry, held in honour of Professor Su Buchin at the centenary of his start, have been held in September 2001 at Fudan college, Shanghai, China. round a hundred mathematicians from China, France, Japan, Singapore and the us participated. This quantity of court cases covers a vast spectrum of complex subject matters in arithmetic, in particular in differential geometry, resembling a few difficulties of universal curiosity in harmonic maps, submanifolds, the Yang-Mills box and the geometric idea of solitons.

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7, and the set Eoo has no such property, but the (n — 4 + /3)-dimensional Hausdorff measure o/Soo is zero for any P > 0. Proof 1. The estimation of the overlapping number of a covering. j, I = 1,2, • • •} such that Ri —> 0 as I —¥ oo. Now we fix Ri. We can choose a family {BR{/2(XU), i = l , - " i-FJ} °f disjoint balls centred at point xu with radius Ri/2, such that the number Pi of these balls attains maximum in the following sense : for any Pi + 1 balls with radius Ri/2, there exist at least two balls which overlap each other.

Let At be a regular Yang-Mills flow with an initial connection AQ for any time t > 0. )-dimensional Hausdorff measure 'H n ~ 4 + / 3 (E 0 0 ) for any (3 > 0. Proof 1. |2<^p on PR{ (xi,tm) (4-20) for any tm. Namely, \FA, | are uniformly bounded on PR; {xi,tm) for all tm- We fix any one of the balls BR{{xi) in this covering, and denote it simply by B = Then \FAtm | are uniformly bounded on B for BR^XI). 1, on any rather small domain Pm in Pm — PR{ {xi,tm), not only \FAt | but also the norms of higher gauge- covariant derivatives IVj^i 7 ^,!

T|2< ° " -*2+Q' 23 §3. 2) where constants Ck are dependent only on k, R, A, YMQ, Co and the geometry of M. Proof 1° Without lossing generality, we suppose (xo,to) = (0,0). We write F = FA = FAt, VA = Vyt, for simplicity. From Bochner type inequality for \FA\2, we have 2 | V ^ F A | 2 < A\FA\2 - ^\FA\2 \FA\2 . 3) Let B = BR(0), B' = BR. (0), T' = {t\ \t\ < (R1)2}, where XR < R! < R. We choose the cut-off function tp on BRM (0 < ip < 1) such that, V = 1 inside B' and ip = 0 outside B. 3) and integrating over B and T", we obtain 2 / I V 2 |VA^| 2 JT> JB £ /, /.