A ''sup+ c inf'' inequality for the equation -delta u = [ by Bartolucci D.

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By Bartolucci D.

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K-i outside a compact set K, its metric can be written as n-l 2 2 ds = dr + J2 9ij(r,0). We define F(s,w) — {(r, 8) € M \sw < r < (s + l)w}, for Ww > 0 and s G Z+, and denote V(s,w) the volume of F(s,w). 12) where the definition of a, r is as same as the above. Proof. 15) )cdtdB J(s+l)w n r(s+2)w < 2/ / »(1) J(s+l)w Jsn-Hi) J(i = 2V(s + l,w) Where we choose s0 such that ( Similarly, we have 8 y/g(t)dtd6 ^~° ) c < 2, so the last inequality is valid. THE P R O O F OF T H E O R E M 1 For simplicity, the discussion in this section is always under the assumption that the integer k is sufficiently large.

Up to now, these limiting connections A^ exist on each good ball individually. 2]), then it becomes a limiting connection over M\Soo. In fact, suppose Bi,Bj are two good balls with above properties, and Bitl Bj ^ 0. , _ s-~ioo Am = °m(Am) >Aoo (») (») (*) (») on Bi. Since Bj is good ball for all tm, so it is also a good ball for the chosen subsequence for Bi. oo on Bj. 5], we can take a sub(0 U) sequence { s m } , such that {sm} converges in C°° to a gauge transformation Soo which takes A^, to AQQ on the overlapping domain.

H} be the balls such that Bv(R,){xilyi) n •••n B^R^X^J) Then, using triangular inequality, B^R^Xi^i) Since {BRl/2{xij,i)} ^ 0. 5) ,ht. 28 2. The definitions of good balls and bad balls, the estimation of the number of bad balls. »)(£CRi))

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