By R. Martini, E.M. de Jager
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Additional info for Geometric Techniques in Gauge Theories
Example text
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 £ /, /.