%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: book.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o book.ps book.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2001.06.12:1514 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 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b(oc)m(k)28 b(decomp)s(osition)d(of)h(the)456 1497 y(Dirac)31 b(Laplacian)1236 1719 y FF(D)1320 1678 y FC(\000)1317 1746 y FH(A)1379 1719 y FF(D)1463 1678 y FB(+)1460 1746 y FH(A)1549 1719 y FM(=)d FG(r)1736 1678 y FC(\003)1736 1743 y FH(A)1793 1719 y FG(r)1876 1734 y FH(A)1955 1719 y FM(+)2063 1651 y(1)p 2063 1696 49 4 v 2063 1787 a(4)2122 1719 y FF(s)2168 1734 y FH(g)2230 1719 y FM(+)2338 1651 y(1)p 2338 1696 V 2338 1787 a(2)2396 1719 y FF(\015)5 b FM(\()p FF(F)2567 1678 y FB(+)2572 1747 y(^)2553 1763 y FH(A)2626 1719 y FM(\))456 1927 y(implies)32 b(strong)j(b)s(ounds)g(on)g(solutions)e(to)i(the)g(monop)s (ole)d(equations.)51 b(These)456 2043 y(b)s(ounds)30 b(sev)m(erely)i(constrain)e(the)g(p)s(ossible)f(solutions,)h(and)g (imply)e(in)h(particu-)456 2159 y(lar)i(that)h(the)h(mo)s(duli)d (spaces)k(are)f(alw)m(a)m(ys)g(compact.)605 2275 y(These)44 b(results)e(ha)m(v)m(e)h(no)f(analogues)f(for)h(most)f(other)h (equations)g(of)g(this)456 2392 y(t)m(yp)s(e,)g(b)s(e)e(they)g(the)g (an)m(ti{self{dual)d(Y)-8 b(ang{Mills)37 b(equations,)42 b(or)d(v)-5 b(arian)m(ts)39 b(of)456 2508 y(the)i(monop)s(ole)e (equations)j(in)m(v)m(olving)e(higher)g(rank)i(connections)g(or)f(sev)m (eral)456 2624 y(spinors,)32 b(or)h(the)g(equations)f(for)g (pseudo{holomorphic)g(curv)m(es.)875 2883 y FD(1.)57 b(Estimates)35 b(from)i(the)g(W)-9 b(eitzen)m(b\177)-56 b(oc)m(k)36 b(form)m(ula)605 3057 y FM(Let)44 b FF(X)52 b FM(b)s(e)44 b(a)g(closed)g(orien)m(ted)g(4{manifold)c(with)k(a)g (\014xed)h(Riemannian)456 3173 y(metric)31 b FF(g)36 b FM(and)d(a)f(\014xed)i(Spin)1538 3132 y FH(c)1573 3173 y FM({structure)2062 3148 y(~)2040 3173 y FF(P)13 b FM(.)605 3290 y(The)36 b(most)e(fundamen)m(tal)g(b)s(ound)h(on)g(solutions)f(\() p FF(A;)17 b(\036)p FM(\))34 b(of)h(the)g(monop)s(ole)456 3406 y(equations)d(\(2.1\))h(and)f(\(2.3\))g(is)h(a)f FF(C)1805 3370 y FB(0)1877 3406 y FM(b)s(ound)h(on)f FF(\036)h FM(obtained)f(from)f(the)i(maxi-)456 3522 y(m)m(um)g (principle.)49 b(Recall)33 b(that)i(all)e(our)h(con\014gurations)h(are) g(of)f(Sob)s(olev)g(class)456 3638 y FF(L)522 3602 y FB(2)522 3663 y(4)561 3638 y FM(,)48 b(and)c(so)g(are)h(con)m(tin)m (uous.)79 b(A)m(t)45 b(a)e(p)s(oin)m(t)h(where)h FG(j)p FF(\036)p FG(j)f FM(ac)m(hiev)m(es)i(its)d(maxi-)456 3755 y(m)m(um,)35 b(w)m(e)i(calculate)d(using)h(the)h(W)-8 b(eitzen)m(b\177)-49 b(oc)m(k)38 b(form)m(ula,)c(the)i(equations)g(and) 456 3871 y(Lemma)31 b(2.3:)703 4073 y(0)d FG(\024)895 4005 y FM(1)p 895 4050 V 895 4141 a(2)954 4073 y(\001)p FG(j)p FF(\036)p FG(j)1149 4032 y FB(2)1215 4073 y FM(=)g FG(hr)1441 4032 y FC(\003)1441 4097 y FH(A)1498 4073 y FG(r)1581 4088 y FH(A)1638 4073 y FF(\036;)17 b(\036)p FG(i)k(\000)h(jr)2068 4088 y FH(A)2125 4073 y FF(\036)p FG(j)2211 4032 y FB(2)2277 4073 y FG(\024)29 b(hr)2505 4032 y FC(\003)2505 4097 y FH(A)2561 4073 y FG(r)2644 4088 y FH(A)2701 4073 y FF(\036;)17 b(\036)p FG(i)780 4310 y FM(=)27 b FG(h)p FF(D)1006 4269 y FC(\000)1003 4337 y FH(A)1065 4310 y FF(D)1149 4269 y FB(+)1146 4337 y FH(A)1208 4310 y FF(\036;)17 b(\036)p FG(i)k(\000)1537 4243 y FM(1)p 1537 4287 V 1537 4378 a(4)1596 4310 y FF(s)1642 4325 y FH(g)1682 4310 y FG(j)p FF(\036)p FG(j)1796 4269 y FB(2)1857 4310 y FG(\000)1966 4243 y FM(1)p 1966 4287 V 1966 4378 a(2)2025 4310 y FG(h)p FF(\015)5 b FM(\()p FF(F)2235 4269 y FB(+)2239 4338 y(^)2221 4355 y FH(A)2293 4310 y FM(\))p FF(\036;)17 b(\036)p FG(i)780 4547 y FM(=)27 b FG(\000)970 4480 y FM(1)p 970 4525 V 970 4616 a(4)1030 4547 y FF(s)1076 4562 y FH(g)1116 4547 y FG(j)p FF(\036)p FG(j)1230 4506 y FB(2)1290 4547 y FG(\000)1400 4480 y FM(1)p 1400 4525 V 1400 4616 a(2)1459 4547 y FG(h)p FF(\015)5 b FM(\()p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\)\))p FF(\036;)g(\036)p FG(i)25 b FM(=)i FG(\000)2339 4480 y FM(1)p 2339 4525 V 2339 4616 a(4)2399 4547 y FF(s)2445 4562 y FH(g)2485 4547 y FG(j)p FF(\036)p FG(j)2599 4506 y FB(2)2659 4547 y FG(\000)c FM(2)p FG(j)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))p FG(j)3159 4506 y FB(2)780 4785 y FM(=)27 b FG(\000)970 4717 y FM(1)p 970 4762 V 970 4853 a(4)1030 4785 y FF(s)1076 4800 y FH(g)1116 4785 y FG(j)p FF(\036)p FG(j)1230 4744 y FB(2)1290 4785 y FG(\000)1400 4717 y FM(1)p 1400 4762 V 1400 4853 a(4)1459 4785 y FG(j)p FF(\036)p FG(j)1573 4744 y FB(4)1644 4785 y FF(:)456 4993 y FM(Th)m(us,)f(either)d FF(\036)g FM(v)-5 b(anishes)23 b(iden)m(tically)e(and)i(all)e(solutions)h(are)h (reducible,)i(or)e(the)456 5109 y(scalar)28 b(curv)-5 b(ature)29 b(function)f FF(s)1585 5124 y FH(g)1653 5109 y FM(is)g(negativ)m(e)h(somewhere)h(and)f FG(j)p FF(\036)p FG(j)2918 5073 y FB(2)2985 5109 y FM(is)f(b)s(ounded)456 5225 y(ab)s(o)m(v)m(e,)33 b(p)s(oin)m(t)m(wise,)g(b)m(y)g(the)g(maxim)m (um)d(of)i FG(\000)p FF(s)2210 5240 y FH(g)2283 5225 y FM(o)m(v)m(er)i FF(X)8 b FM(.)1912 5375 y FE(39)p eop %%Page: 40 2 40 39 bop 456 235 a FE(40)346 b(3.)35 b(BOUNDS)c(ON)i(SOLUTIONS)e(AND)i (THEIR)f(APPLICA)-6 b(TIONS)605 435 y FM(Notice)37 b(that,)h(using)f (Lemma)f(2.3)g(again,)h(the)h(curv)-5 b(ature)38 b(equation)f(\(2.3\)) 456 551 y(implies)1342 720 y FG(j)p FF(F)1447 679 y FB(+)1452 748 y(^)1433 765 y FH(A)1506 720 y FG(j)27 b FM(=)h FG(j)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))p FG(j)26 b FM(=)2221 653 y(1)p 2155 697 181 4 v 2155 799 a(2)2204 717 y FG(p)p 2287 717 49 4 v 82 x FM(2)2346 720 y FG(j)p FF(\036)p FG(j)2460 679 y FB(2)2531 720 y FF(;)456 932 y FM(so)32 b(that)h(the)g FF(C)1032 896 y FB(0)1104 932 y FM(b)s(ound)f(on)h FF(\036)f FM(also)g(giv)m(es)h(a)f FF(C)2223 896 y FB(0)2295 932 y FM(b)s(ound)h(on)f FF(F)2808 891 y FB(+)2813 960 y(^)2794 977 y FH(A)2867 932 y FM(.)605 1054 y(It)24 b(is)f(an)g(easy)h(exercise)h(to)e(carry)h(out)g(the)f(ab) s(o)m(v)m(e)i(calculation)c(with)i(the)h(p)s(er-)456 1171 y(turb)s(ed)35 b(curv)-5 b(ature)35 b(equation)f(\(2.5\))g (instead)h(of)f(the)h(unp)s(erturb)s(ed)h(one)f(\(2.3\))o(.)456 1287 y(One)40 b(\014nds)h(that)e(for)h(a)f(solution)g(\()p FF(A;)17 b(\036)p FM(\))39 b(of)g(equations)h(\(2.4\))g(and)g(\(2.5\))o (,)i(the)456 1403 y(follo)m(wing)29 b(estimate)j(holds)g(at)h(a)f(p)s (oin)m(t)g(where)i FG(j)p FF(\036)p FG(j)d FM(is)h(maximal:)819 1618 y(0)27 b FG(\024)1010 1551 y FM(1)p 1010 1595 V 1010 1687 a(2)1069 1618 y(\001)p FG(j)p FF(\036)p FG(j)1264 1577 y FB(2)1330 1618 y FG(\024)i(h)p FF(D)1559 1577 y FC(\000)1556 1645 y FH(A)1617 1618 y FF(D)1701 1577 y FB(+)1698 1645 y FH(A)1760 1618 y FF(\036;)17 b(\036)p FG(i)k(\000)2089 1551 y FM(1)p 2089 1595 V 2089 1687 a(4)2148 1618 y FF(s)2194 1633 y FH(g)2234 1618 y FG(j)p FF(\036)p FG(j)2348 1577 y FB(2)2409 1618 y FG(\000)2518 1551 y FM(1)p 2518 1595 V 2518 1687 a(2)2577 1618 y FG(h)p FF(\015)5 b FM(\()p FF(F)2787 1577 y FB(+)2792 1646 y(^)2773 1663 y FH(A)2846 1618 y FM(\))p FF(\036;)17 b(\036)p FG(i)1330 1856 y FM(=)28 b FG(\000)1521 1788 y FM(1)p 1521 1833 V 1521 1924 a(4)1580 1856 y FF(s)1626 1871 y FH(g)1666 1856 y FG(j)p FF(\036)p FG(j)1780 1814 y FB(2)1841 1856 y FG(\000)1950 1788 y FM(1)p 1950 1833 V 1950 1924 a(2)2009 1856 y FG(h)p FF(\015)5 b FM(\()p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))k(+)h FF(!)t FM(\))p FF(\036;)17 b(\036)p FG(i)1330 2093 y FM(=)28 b FG(\000)1521 2025 y FM(1)p 1521 2070 V 1521 2161 a(4)1580 2093 y FF(s)1626 2108 y FH(g)1666 2093 y FG(j)p FF(\036)p FG(j)1780 2052 y FB(2)1841 2093 y FG(\000)1950 2025 y FM(1)p 1950 2070 V 1950 2161 a(4)2009 2093 y FG(j)p FF(\036)p FG(j)2123 2052 y FB(4)2184 2093 y FG(\000)22 b FM(2)p FG(h)p FF(!)t(;)17 b(\033)t FM(\()p FF(\036;)g(\036)p FM(\))p FG(i)1330 2330 y(\024)29 b(\000)1523 2263 y FM(1)p 1523 2307 V 1523 2399 a(4)1582 2330 y FF(s)1628 2345 y FH(g)1668 2330 y FG(j)p FF(\036)p FG(j)1782 2289 y FB(2)1842 2330 y FG(\000)1952 2263 y FM(1)p 1952 2307 V 1952 2399 a(4)2011 2330 y FG(j)p FF(\036)p FG(j)2125 2289 y FB(4)2185 2330 y FM(+)22 b(2)p FG(j)p FF(!)t FG(j)f(\001)h(j)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))p FG(j)1330 2567 y FM(=)28 b FG(\000)1521 2500 y FM(1)p 1521 2545 V 1521 2636 a(4)1580 2567 y FF(s)1626 2582 y FH(g)1666 2567 y FG(j)p FF(\036)p FG(j)1780 2526 y FB(2)1841 2567 y FG(\000)1950 2500 y FM(1)p 1950 2545 V 1950 2636 a(4)2009 2567 y FG(j)p FF(\036)p FG(j)2123 2526 y FB(4)2184 2567 y FM(+)2333 2500 y(1)p 2292 2545 132 4 v 2292 2564 a FG(p)p 2375 2564 49 4 v 83 x FM(2)2434 2567 y FG(j)p FF(!)t FG(j)20 b(\001)i(j)p FF(\036)p FG(j)2739 2526 y FB(2)2810 2567 y FF(:)456 2800 y FM(As)33 b(ab)s(o)m(v)m(e,)g(this)f (also)g(giv)m(es)h(a)f(b)s(ound)h(on)g FF(F)2123 2759 y FB(+)2127 2828 y(^)2109 2845 y FH(A)2181 2800 y FM(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(pro)m(v)m(ed:)605 3007 y FL(Pr)n(oposition)k(3.1.)48 b FK(L)-5 b(et)34 b FF(s)1681 3022 y FH(g)r(;!)1815 3007 y FM(=)27 b FK(min)p FG(f)p FM(0)p FF(;)50 b(s)2305 3022 y FH(g)2364 3007 y FG(\000)20 b FM(2)2510 2925 y FG(p)p 2592 2925 V 2592 3007 a FM(2)p FG(j)p FF(!)t FG(jg)32 b FK(with)i(the)f(mini-)456 3123 y(mum)h(taken)g(over)g(al)5 b(l)34 b(of)h FF(X)8 b FK(,)34 b(and)g(let)h FM(\()p FF(A;)17 b(\036)p FM(\))34 b FK(b)-5 b(e)34 b(a)g(solution)h(to)f(the)h(monop)-5 b(ole)456 3240 y(e)g(quations)49 b FM(\(2.4\))40 b FK(and)51 b FM(\(2.5\))p FK(.)64 b(Then)40 b FF(\036)h FK(and)g FF(F)2273 3198 y FB(+)2278 3267 y(^)2259 3284 y FH(A)2373 3240 y FK(ar)-5 b(e)41 b(p)-5 b(ointwise)40 b(b)-5 b(ounde)g(d)41 b(as)456 3362 y(fol)5 b(lows:)1374 3528 y FG(j)p FF(\036)p FG(j)1488 3487 y FB(2)1554 3528 y FG(\024)28 b(\000)p FF(s)1782 3543 y FH(g)r(;!)1923 3528 y FF(;)-1494 b FM(\(3.1\))1335 3736 y FG(j)p FF(F)1440 3695 y FB(+)1444 3764 y(^)1426 3781 y FH(A)1498 3736 y FG(j)28 b(\024)g(\000)1812 3669 y FM(1)p 1746 3713 181 4 v 1746 3816 a(2)1795 3733 y FG(p)p 1878 3733 49 4 v 83 x FM(2)1937 3736 y FF(s)1983 3751 y FH(g)r(;!)2111 3736 y FM(+)22 b FK(max)o FG(j)p FF(!)t FG(j)34 b FF(:)-2109 b FM(\(3.2\))605 3980 y(These)34 b(b)s(ounds)g(are)e(go)s(o)s(d)g(as)g FF(!)k FM(is)c(in)g FF(L)2122 3944 y FB(2)2122 4004 y(3)2195 3980 y FM(and)g(therefore)h (con)m(tin)m(uous.)605 4096 y(In)38 b(addition)e(to)i(these)h(p)s(oin)m (t)m(wise)f(b)s(ounds,)i(w)m(e)f(also)e(ha)m(v)m(e)i(the)f(follo)m (wing)456 4212 y(in)m(tegral)31 b(b)s(ounds)i(on)f(all)f(solutions:)605 4392 y FL(Pr)n(oposition)38 b(3.2.)48 b FK(F)-7 b(or)27 b(every)g(solution)h FM(\()p FF(A;)17 b(\036)p FM(\))26 b FK(to)i(the)g(monop)-5 b(ole)26 b(e)-5 b(qua-)456 4508 y(tions)42 b FM(\(2.4\))35 b FK(and)44 b FM(\(2.5\))35 b FK(we)f(have:)1075 4687 y FG(jj)p FF(\036)p FG(jj)1245 4646 y FB(2)1245 4713 y FH(L)1293 4694 y Fy(4)1358 4687 y FG(\024)28 b(jj)p FM(\()p FG(\000)p FF(s)1680 4702 y FH(g)1742 4687 y FM(+)22 b(2)1889 4600 y FG(p)p 1972 4600 V 87 x FM(2)p FG(j)p FF(!)t FG(j)p FM(\))p FG(jj)2236 4704 y FH(L)2284 4685 y Fy(2)2355 4687 y FF(;)-1926 b FM(\(3.3\))935 4895 y FG(jjr)1074 4910 y FH(A)1131 4895 y FF(\036)p FG(jj)1245 4912 y FH(L)1293 4893 y Fy(2)1358 4895 y FG(\024)1473 4828 y FM(1)p 1473 4872 V 1473 4964 a(2)1532 4895 y FG(jj)p FM(\()p FG(\000)p FF(s)1749 4910 y FH(g)1811 4895 y FM(+)22 b(2)1958 4808 y FG(p)p 2041 4808 V 87 x FM(2)o FG(j)p FF(!)t FG(j)p FM(\))p FG(jj)2304 4912 y FH(L)2352 4893 y Fy(2)2424 4895 y FF(;)-1995 b FM(\(3.4\))997 5132 y FG(jj)p FF(F)1130 5091 y FB(+)1134 5160 y(^)1116 5177 y FH(A)1188 5132 y FG(jj)1244 5149 y FH(L)1292 5130 y Fy(2)1358 5132 y FG(\024)1539 5065 y FM(1)p 1473 5110 181 4 v 1473 5212 a(2)1522 5129 y FG(p)p 1605 5129 49 4 v 83 x FM(2)1664 5132 y FG(jj)p FM(\()p FG(\000)p FF(s)1881 5147 y FH(g)1943 5132 y FM(+)22 b(2)2090 5045 y FG(p)p 2172 5045 V 2172 5132 a FM(2)p FG(j)p FF(!)t FG(j)p FM(\))p FG(jj)2436 5149 y FH(L)2484 5130 y Fy(2)2543 5132 y FM(+)g FG(jj)p FF(!)t FG(jj)2818 5149 y FH(L)2866 5130 y Fy(2)2938 5132 y FF(:)-2509 b FM(\(3.5\))p eop %%Page: 41 3 41 40 bop 925 253 a FE(1.)35 b(ESTIMA)-6 b(TES)33 b(FR)n(OM)g(THE)h (WEITZENB)2405 234 y(\177)2394 253 y(OCK)g(F)n(ORMULA)366 b(41)605 435 y FL(Pr)n(oof.)42 b FM(W)-8 b(e)31 b(apply)g(b)s(oth)g (sides)g(of)g(the)g(W)-8 b(eitzen)m(b\177)-49 b(oc)m(k)33 b(form)m(ula)c(to)i FF(\036)f FM(and)456 551 y(tak)m(e)39 b(the)g FF(L)913 515 y FB(2)992 551 y FM(inner)f(pro)s(duct)h(with)f FF(\036)p FM(.)62 b(After)38 b(substituting)h(from)e(the)i(equa-)456 667 y(tions,)32 b(w)m(e)h(obtain:)867 897 y(0)28 b(=)1047 762 y Fw(Z)1103 987 y FH(X)1170 897 y FM(\()p FG(jr)1319 912 y FH(A)1376 897 y FF(\036)p FG(j)1462 856 y FB(2)1523 897 y FM(+)1631 830 y(1)p 1631 875 49 4 v 1631 966 a(4)1690 897 y FF(s)1736 912 y FH(g)1776 897 y FG(j)p FF(\036)p FG(j)1890 856 y FB(2)1950 897 y FM(+)2058 830 y(1)p 2058 875 V 2058 966 a(4)2117 897 y FG(j)p FF(\036)p FG(j)2231 856 y FB(4)2292 897 y FM(+)22 b(2)p FG(h)p FF(!)t(;)17 b(\033)t FM(\()p FF(\036;)g(\036)p FM(\))p FG(i)p FM(\))p FF(dv)t(ol)3137 912 y FH(g)3206 897 y FF(:)-2777 b FM(\(3.6\))456 1132 y(This)32 b(sho)m(ws)658 1207 y Fw(Z)714 1433 y FH(X)798 1343 y FG(j)p FF(\036)p FG(j)912 1302 y FB(4)950 1343 y FF(dv)t(ol)1128 1358 y FH(g)1195 1343 y FG(\024)1301 1207 y Fw(Z)1356 1433 y FH(X)1423 1343 y FM(\()p FG(\000)p FF(s)1584 1358 y FH(g)1625 1343 y FG(j)p FF(\036)p 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FM(\()2050 1996 y Fw(Z)2105 2221 y FH(X)2173 2132 y FM(\()p FG(\000)p FF(s)2334 2147 y FH(g)2396 2132 y FM(+)g(2)2543 2044 y FG(p)p 2626 2044 49 4 v 88 x FM(2)p FG(j)p FF(!)t FG(j)p FM(\))2834 2090 y FB(2)2872 2132 y FF(dv)t(ol)3050 2147 y FH(g)3090 2132 y FM(\))3138 2063 y Fy(1)p 3137 2075 31 3 v 3137 2116 a(2)3215 2132 y FF(;)456 2371 y FM(whic)m(h)33 b(implies)d(\(3.3\))o(.)605 2488 y(Starting)42 b(again)f(with)h (\(3.6\),)j(pro)s(ceeding)e(as)g(ab)s(o)m(v)m(e)g(and)g(using)f(the)i (in-)456 2604 y(equalit)m(y)32 b(w)m(e)h(just)g(pro)m(v)m(ed,)i (\(3.4\))d(follo)m(ws.)605 2720 y(T)-8 b(o)33 b(pro)m(v)m(e)h(\(3.5\))e (w)m(e)h(calculate)860 2820 y Fw(Z)915 3046 y FH(X)999 2956 y FG(j)p FF(F)1104 2915 y FB(+)1108 2984 y(^)1090 3000 y FH(A)1162 2956 y FG(j)1190 2915 y FB(2)1229 2956 y FF(dv)t(ol)1407 2971 y FH(g)1475 2956 y FM(=)1578 2820 y Fw(Z)1634 3046 y FH(X)1717 2956 y FG(j)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))k(+)h FF(!)t FG(j)2252 2915 y FB(2)2291 2956 y FF(dv)t(ol)2469 2971 y FH(g)1475 3219 y FG(\024)1580 3083 y Fw(Z)1635 3309 y FH(X)1703 3219 y FM(\()1751 3151 y(1)p 1751 3196 49 4 v 1751 3287 a(8)1809 3219 y FG(j)p FF(\036)p FG(j)1923 3178 y FB(4)1984 3219 y FM(+)2133 3151 y(1)p 2092 3196 132 4 v 2092 3216 a FG(p)p 2175 3216 49 4 v 82 x FM(2)2234 3219 y FG(j)p FF(\036)p FG(j)2348 3178 y FB(2)2386 3219 y FG(j)p FF(!)t FG(j)f FM(+)h FG(j)p FF(!)t FG(j)2747 3178 y FB(2)2785 3219 y FM(\))p FF(dv)t(ol)3001 3234 y FH(g)1475 3484 y FM(=)1578 3349 y Fw(Z)1634 3574 y FH(X)1701 3484 y FM(\()1815 3417 y(1)p 1749 3462 181 4 v 1749 3564 a(2)1798 3481 y FG(p)p 1881 3481 49 4 v 83 x FM(2)1939 3484 y FG(j)p FF(\036)p FG(j)2053 3443 y FB(2)2114 3484 y FM(+)g FG(j)p FF(!)t FG(j)p FM(\))2371 3443 y FB(2)2409 3484 y FF(dv)t(ol)2587 3499 y FH(g)2659 3484 y FF(:)456 3727 y FM(No)m(w)33 b(the)g(claim)d(follo)m(ws)h(from)h(\(3.3\))o(.)p 3374 3727 4 66 v 3378 3665 59 4 v 3378 3727 V 3436 3727 4 66 v 605 3917 a FL(Remark)38 b(3.3.)48 b FM(Sp)s(ecialising)30 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b(!)t FM(\))26 b FG(2)i(P)41 b FK(ther)-5 b(e)32 b(ar)-5 b(e)32 b(at)h(most)f(\014nitely)g(many)g FM(Spin)2944 4951 y FH(c)2979 4993 y FK({structur)-5 b(es)456 5109 y(on)31 b FF(X)39 b FK(for)32 b(which)e FF(M)1229 5124 y FH(!)1312 5109 y FK(is)h(not)h(empty,)g(and)f(of)h(non{ne)-5 b(gative)30 b(exp)-5 b(e)g(cte)g(d)30 b(dimen-)456 5225 y(sion.)p eop %%Page: 42 4 42 41 bop 456 235 a FE(42)346 b(3.)35 b(BOUNDS)c(ON)i(SOLUTIONS)e(AND)i (THEIR)f(APPLICA)-6 b(TIONS)605 435 y FL(Pr)n(oof.)42 b FM(The)23 b(assumption)e(of)g(non{negativ)m(e)h(exp)s(ected)i (dimension)c(means)456 551 y(that)1384 675 y FF(c)1426 634 y FB(2)1426 700 y(1)1465 675 y FM(\()p FF(V)1560 690 y FC(\006)1619 675 y FM(\))27 b FG(\025)i FM(2)p FF(\037)p FM(\()p FF(X)8 b FM(\))21 b(+)h(3)p FF(\033)t FM(\()p FF(X)8 b FM(\))32 b FF(:)456 820 y FM(Th)m(us)c FF(c)739 784 y FB(2)739 845 y(1)779 820 y FM(\()p FF(V)874 835 y FC(\006)933 820 y FM(\))f(is)f(b)s(ounded)i(b)s(elo)m(w,)h(and)e (the)h(Chern{W)-8 b(eil)26 b(form)m(ula)g 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FM(\))48 b FF(>)h FM(0)p FK(,)g(and)c(that)i FM(\()p FF(g)t(;)17 b(!)t FM(\))47 b FG(2)i(P)456 2159 y FK(is)39 b(generic.)56 b(Then)39 b(ther)-5 b(e)39 b(ar)-5 b(e)39 b(at)g(most)g(\014nitely)g (many)g FM(Spin)2784 2117 y FH(c)2819 2159 y FK({structur)-5 b(es)40 b(for)456 2275 y(which)34 b FF(M)824 2290 y FH(!)909 2275 y FK(is)h(nonempty.)605 2455 y FL(Pr)n(oof.)42 b FM(By)31 b(the)h(transv)m(ersalit)m(y)f(result)g(of)f(Section)h(6)f(in) g(Chapter)i(2,)f(the)456 2571 y(assumption)k FF(b)1016 2530 y FB(+)1016 2595 y(2)1108 2571 y FF(>)e FM(0)j(implies)d(that)j (for)f(generic)h(parameters)g(there)g(are)g(only)456 2687 y(irreducible)j(solutions,)i(and)g(these)h(solutions)d(are)i (transv)m(erse)h(zero)s(es)g(of)e FF(f)3367 2702 y FH(!)3417 2687 y FM(,)456 2803 y(so)27 b(that)h(the)g(only)f(non{empt)m(y)h(mo)s (duli)d(spaces)k(ha)m(v)m(e)g(non{negativ)m(e)e(exp)s(ected)456 2919 y(dimension.)p 3374 2919 V 3378 2857 59 4 v 3378 2919 V 3436 2919 4 66 v 1058 3140 a FD(2.)56 b(Compactness)38 b(of)g(the)f(mo)s(duli)f(space)605 3315 y FM(W)-8 b(e)43 b(shall)e(no)m(w)i(sho)m(w)h(that)e(the)h(mo)s(duli)d(space)k FF(M)53 b FM(of)42 b(solutions)g(to)g(the)456 3431 y(monop)s(ole)25 b(equations)j(mo)s(dulo)e(gauge)h(equiv)-5 b(alence)28 b(is)f(compact.)41 b(The)29 b(idea)e(of)456 3547 y(the)22 b(pro)s(of)f(is)h(rather)g(simple,)h(and)f(is)f(a)h(routine)g (application)d(of)j(the)g(inequalities)456 3664 y(pro)m(v)m(ed)34 b(in)d(the)i(previous)g(Section.)605 3780 y(W)-8 b(e)44 b(\014x)g(a)f(closed)g(orien)m(ted)h(Riemannian)d(four{manifold)f(\()p FF(X)r(;)17 b(g)t FM(\))42 b(and)i(a)456 3896 y(Spin)645 3854 y FH(c)718 3896 y FM(structure)c(on)f(it.)60 b(F)-8 b(or)37 b(simplicit)m(y)f(w)m(e)k(shall)d(\014rst)i(discuss)g(the)g (unp)s(er-)456 4012 y(turb)s(ed)33 b(equations)f(\(2.1\))g(and)h (\(2.3\).)605 4128 y(Let)26 b(\()p FF(A;)17 b(\036)p FM(\))26 b(b)s(e)g(a)g(solution)e(of)i(equations)g(\(2.1\))g(and)g (\(2.3\),)h(and)f(let)g FF(A)3204 4143 y FB(0)3269 4128 y FM(b)s(e)h(a)456 4245 y(\014xed)j(reference)h(connection.)43 b(Recall)28 b(that)h(\(3.5\))g(giv)m(es)h(an)f FF(L)2782 4209 y FB(2)2852 4245 y FM(b)s(ound)g(on)h FF(F)3359 4203 y FB(+)3363 4272 y(^)3345 4289 y FH(A)3417 4245 y FM(,)456 4376 y(and)38 b(that)f(this)h(together)g(with)f(the)h (Chern{W)-8 b(eil)38 b(form)m(ula)e(also)h(giv)m(es)h(an)g FF(L)3405 4340 y FB(2)456 4499 y FM(b)s(ound)33 b(on)h FF(F)971 4457 y FC(\000)976 4526 y FB(^)957 4543 y FH(A)1030 4499 y FM(,)g(so)g(that)f(the)h(total)e(curv)-5 b(ature)35 b(of)2402 4473 y(^)2377 4499 y FF(A)e FM(is)h(b)s(ounded)g(in)f FF(L)3163 4462 y FB(2)3203 4499 y FM(.)46 b(The)456 4626 y(map)33 b(from)h FG(A)p FF(=)p FG(G)40 b FM(to)34 b(curv)-5 b(ature)35 b(t)m(w)m(o{forms)f(of)g(the)h(induced)g(connections)g(on) 456 4742 y FF(det)587 4757 y Fv(C)639 4742 y FM(\()p FF(V)734 4757 y FC(\006)793 4742 y FM(\))f(has)h(compact)f(\014b)s(ers) h(isomorphic)d(to)i FF(J)9 b FM(\()p FF(X)f FM(\).)48 b(Th)m(us)36 b(the)f FF(L)3102 4706 y FB(2)3176 4742 y FM(b)s(ound)456 4870 y(on)d(the)h(curv)-5 b(ature)33 b(giv)m(es)g(an)g FF(L)1636 4834 y FB(2)1708 4870 y FM(b)s(ound)g(on) 2170 4845 y(^)2144 4870 y FF(A)q FM(,)f(and)h(the)g(pro)5 b(jection)1628 5035 y FF(\031)15 b FM(:)33 b FF(M)39 b FG(\000)-17 b(!)28 b(A)p FF(=)p FG(G)1634 5186 y FM([)p FF(A;)17 b(\036)p FM(])28 b FG(7\000)-17 b(!)28 b FM([)p FF(A)p FM(])p eop %%Page: 43 5 43 42 bop 1128 235 a FE(2.)35 b(COMP)-6 b(A)n(CTNESS)33 b(OF)h(THE)f(MODULI)g(SP)-6 b(A)n(CE)570 b(43)456 435 y FM(is)32 b(b)s(ounded)h(in)f FF(L)1132 399 y FB(2)1172 435 y FM(.)605 551 y(By)37 b(applying)f(gauge)g(transformations)g(in)g (the)h(comp)s(onen)m(t)g(of)f(the)h(iden-)456 667 y(tit)m(y)-8 b(,)32 b(w)m(e)h(can)g(ac)m(hiev)m(e)1593 853 y FF(d)1644 812 y FC(\003)1684 853 y FM(\()p FF(A)22 b FG(\000)g FF(A)1989 868 y FB(0)2029 853 y FM(\))28 b(=)f(0)32 b FF(:)456 1060 y FM(Th)m(us)48 b FF(A)32 b FG(\000)g FF(A)1004 1075 y FB(0)1096 1060 y FM(=)52 b FF(h)32 b FM(+)g FF(\013)q FM(,)50 b(with)d FF(h)g FM(harmonic)e(and)i FF(\013)53 b FM(=)f FF(d)2840 1023 y FC(\003)2879 1060 y FF(\014)6 b FM(.)86 b(Using)46 b(all)456 1176 y(comp)s(onen)m(ts)f(of)f(the)i (gauge)e(group,)k(w)m(e)e(ma)m(y)f(assume)g(that)g(the)g(harmonic)456 1292 y(part)32 b FF(h)h FM(is)f(b)s(ounded)h(\(in)f(an)m(y)h(norm\),)f (b)s(ecause)i FF(J)9 b FM(\()p FF(X)f FM(\))32 b(is)g(compact.)605 1408 y(As)h FF(\013)f FM(is)g(orthogonal)e(to)i(the)h(k)m(ernel)f(of)g (the)h(elliptic)c(op)s(erator)i FF(d)3042 1372 y FC(\003)3103 1408 y FM(+)21 b FF(d)3251 1372 y FB(+)3341 1408 y FM(on)456 1525 y(1{forms,)31 b(there)i(exists)h(a)e(constan)m(t)h FF(c)g FM(suc)m(h)h(that)456 1866 y FG(jj)p FF(\013)q FG(jj)631 1887 y FH(L)679 1864 y Fy(2)679 1908 y(1)743 1866 y FG(\024)29 b FF(c)p FG(jj)p FM(\()p FF(d)1036 1825 y FC(\003)1096 1866 y FM(+)22 b FF(d)1245 1825 y FB(+)1304 1866 y FM(\))p FF(\013)q FG(jj)1461 1883 y FH(L)1509 1864 y Fy(2)1574 1866 y FM(=)28 b FF(c)p FG(jj)p FF(d)1827 1825 y FB(+)1885 1866 y FF(\013)q FG(jj)2004 1883 y FH(L)2052 1864 y Fy(2)2117 1866 y FM(=)g FF(c)p FG(jj)p FF(F)2396 1825 y FB(+)2400 1894 y(^)2382 1911 y FH(A)2476 1866 y FG(\000)23 b FF(F)2653 1825 y FB(+)2657 1894 y(^)2639 1911 y FH(A)2692 1920 y Fy(0)2730 1866 y FG(jj)2786 1883 y FH(L)2834 1864 y Fy(2)2900 1866 y FG(\024)28 b FF(c)p FG(jj)p FF(F)3180 1825 y FB(+)3184 1894 y(^)3166 1911 y FH(A)3238 1866 y FG(jj)3294 1883 y FH(L)3342 1864 y Fy(2)3403 1866 y FM(+)22 b FF(c)3543 1825 y FC(0)3598 1866 y FF(:)456 1737 y FM(\(3.7\))456 2118 y(Th)m(us)42 b FF(A)28 b FG(\000)g FF(A)990 2133 y FB(0)1072 2118 y FM(=)41 b FF(h)28 b FM(+)g FF(\013)41 b FM(is)g(b)s(ounded)g(in)f FF(L)2181 2081 y FB(2)2181 2142 y(1)2221 2118 y FM(.)69 b(By)41 b(the)g(Rellic)m(h)e(lemma)g(the) 456 2234 y(inclusion)31 b FF(L)929 2198 y FB(2)929 2258 y(1)996 2234 y FG(!)c FF(L)1189 2198 y FB(2)1262 2234 y FM(is)32 b(compact.)43 b(Th)m(us)34 b FF(\031)i FM(is)d(compact)f(in) g(the)h FF(L)2963 2198 y FB(2)3035 2234 y FM(top)s(ology)-8 b(.)605 2350 y(The)31 b(\014b)s(er)f(of)g FF(\031)k FM(o)m(v)m(er)d(a)e (p)s(oin)m(t)h([)p FF(A)p FM(])g(is)f(con)m(tained)h(in)g(the)g (quotien)m(t)g(b)m(y)h FF(U)10 b FM(\(1\))456 2466 y(of)39 b(a)g(subset)j(in)c FF(k)s(er)s FM(\()p FF(D)1357 2425 y FB(+)1354 2493 y FH(A)1416 2466 y FM(\).)65 b(This)40 b(v)m(ector)h(space)f(is)g(\014nite{dimensional,)e(and)456 2582 y(the)d(relev)-5 b(an)m(t)34 b(subset)i(is)e(b)s(ounded)i(b)s (ecause)f(of)g(the)g FF(C)2521 2546 y FB(0)2594 2582 y FM(estimate)f(\(3.1\))g(on)h FF(\036)p FM(.)456 2699 y(Th)m(us)e FF(\031)j FM(has)c(compact)f(\014b)s(ers.)45 b(W)-8 b(e)32 b(conclude)g(that)g FF(M)42 b FM(is)32 b(compact)f(in)g(the)i FF(L)3405 2663 y FB(2)456 2815 y FM(top)s(ology)-8 b(.)605 2931 y(W)g(e)23 b(shall)e(no)m(w)j(pro)m(v) m(e)g(a)e(more)g(precise)i(result,)g(in)e(whic)m(h)i(all)c(the)j(deriv) -5 b(ativ)m(es)456 3047 y(of)31 b(solutions)g(are)h(con)m(trolled)f(as) h(w)m(ell.)43 b(F)-8 b(or)31 b(this)h(w)m(e)h(need)g(to)f(mak)m(e)g (use)h(again)456 3164 y(of)h(the)i(Sob)s(olev)e(m)m(ultiplication)d (theorem)k([)p FD(26)p FM(],)h(whic)m(h)f(w)m(e)i(already)d(recalled) 456 3280 y(and)e(used)i(in)e(Section)g(2)h(of)f(Chapter)h(2.)605 3494 y FL(Theorem)38 b(3.6.)49 b FK(L)-5 b(et)47 b FM(\()p FF(A)1605 3509 y FH(i)1634 3494 y FF(;)17 b(\036)1736 3509 y FH(i)1763 3494 y FM(\))47 b FK(b)-5 b(e)48 b(a)f(se)-5 b(quenc)g(e)46 b(of)i FF(L)2680 3458 y FB(2)2680 3519 y(4)2767 3494 y FK(solutions)f(to)g(the)456 3610 y(monop)-5 b(ole)37 b(e)-5 b(quations)46 b FM(\(2.1\))39 b FK(and)48 b FM(\(2.3\))p FK(.)56 b(Ther)-5 b(e)39 b(is)f(a)h(se)-5 b(quenc)g(e)38 b(of)g FF(L)3129 3574 y FB(2)3129 3635 y(5)3208 3610 y FK(gauge)456 3727 y(tr)-5 b(ansformations)33 b FF(u)1216 3742 y FH(i)1277 3727 y FK(such)h(that)g(the)g FF(u)1913 3742 y FH(i)1941 3727 y FM(\()p FF(A)2052 3742 y FH(i)2080 3727 y FF(;)17 b(\036)2182 3742 y FH(i)2210 3727 y FM(\))34 b FK(ar)-5 b(e)34 b(smo)-5 b(oth)33 b(and)g(have)h(a)g (sub-)456 3843 y(se)-5 b(quenc)g(e)34 b(c)-5 b(onver)g(ging)33 b(in)i(the)g FF(C)1690 3807 y FC(1)1799 3843 y FK(top)-5 b(olo)g(gy)35 b(to)g(a)f FF(C)2451 3807 y FC(1)2561 3843 y FK(solution)h FM(\()p FF(A;)17 b(\036)p FM(\))p FK(.)605 3959 y(In)34 b(p)-5 b(articular,)35 b FF(M)46 b FK(is)34 b(c)-5 b(omp)g(act)34 b(in)h(the)g FF(C)2187 3923 y FC(1)2296 3959 y FK(top)-5 b(olo)g(gy.)605 4173 y FL(Pr)n(oof.)42 b FM(As)48 b(ab)s(o)m(v)m(e,)k(after)c(applying)e(gauge)i (transformations)d(w)m(e)k(can)456 4289 y(write)33 b FF(A)c FM(=)g FF(A)986 4304 y FB(0)1048 4289 y FM(+)22 b FF(h)h FM(+)g FF(\013)q FM(,)33 b(with)g FF(A)1743 4304 y FB(0)1816 4289 y FM(\014xed,)i FF(h)e FM(harmonic,)f(and)i FF(\013)g FM(in)e(the)i(image)456 4406 y(of)j FF(d)623 4370 y FC(\003)663 4406 y FM(;)k(and)d(similarly)d(for)j FF(A)1559 4421 y FH(i)1624 4406 y FM(=)g FF(A)1811 4421 y FB(0)1876 4406 y FM(+)26 b FF(h)2034 4421 y FH(i)2089 4406 y FM(+)f FF(\013)2252 4421 y FH(i)2281 4406 y FM(.)60 b(The)40 b FF(h)2631 4421 y FH(i)2697 4406 y FM(are)f(solutions)e(of)h (a)456 4522 y(linear)32 b(elliptic)f(equation,)k(and)f(w)m(e)h(ma)m(y)f (assume)h(that)f(they)h(are)f(b)s(ounded)h(in)456 4638 y FF(C)533 4602 y FC(1)607 4638 y FM(.)44 b(Th)m(us)34 b(they)f(ha)m(v)m(e)h(a)f(con)m(v)m(ergen)m(t)h(subsequence.)605 4754 y(Recall)39 b(that)i(for)f(ev)m(ery)i(solution)d(\()p FF(A;)17 b(\036)p FM(\),)42 b(w)m(e)g(ha)m(v)m(e)g(a)e(priori)f(b)s (ounds)i(on)456 4871 y FG(jj)p FF(\036)p FG(jj)626 4887 y FH(L)674 4868 y Fy(4)711 4871 y FM(,)c(on)f FG(jjr)1053 4886 y FH(A)1110 4871 y FF(\036)p FG(jj)1224 4887 y FH(L)1272 4868 y Fy(2)1309 4871 y FM(,)i(on)e FG(jj)p FF(\036)p FG(jj)1683 4887 y FH(C)1738 4868 y Fy(0)1775 4871 y FM(,)h(and)f(on)h FG(jj)p FF(F)2305 4829 y FC(\006)2309 4898 y FB(^)2291 4915 y FH(A)2363 4871 y FG(jj)2419 4887 y FH(L)2467 4868 y Fy(2)2505 4871 y FM(.)54 b(When)37 b(b)s(ounding)f(the)456 4993 y(deriv)-5 b(ativ)m(es,)42 b(it)d(will)f(b)s(e)i(imp)s(ortan)m(t)f (to)h(k)m(eep)h(in)f(mind)f(that)h(the)g(norms)g(are)456 5109 y(de\014ned)29 b(with)e(resp)s(ect)i(to)e(a)h(\014xed)h(bac)m (kground)g(connection,)g(sa)m(y)f FF(A)3023 5124 y FB(0)3063 5109 y FM(,)h(and)e(not)456 5225 y(with)32 b(resp)s(ect)i(to)e(the)h(v) -5 b(ariable)31 b FF(A)p FM(.)p eop %%Page: 44 6 44 43 bop 456 235 a FE(44)346 b(3.)35 b(BOUNDS)c(ON)i(SOLUTIONS)e(AND)i (THEIR)f(APPLICA)-6 b(TIONS)605 435 y FM(As)33 b(in)f(\(3.7\))o(,)h(w)m (e)h(ha)m(v)m(e)456 716 y FG(jj)p FF(\013)q FG(jj)631 736 y FH(L)679 703 y Fx(p)679 759 y(k)q Fy(+1)824 716 y FG(\024)28 b FF(c)p FG(jj)p FM(\()p FF(d)1116 675 y FC(\003)1177 716 y FM(+)22 b FF(d)1326 675 y FB(+)1384 716 y FM(\))p FF(\013)q FG(jj)1541 736 y FH(L)1589 703 y Fx(p)1589 759 y(k)1658 716 y FM(=)28 b FF(c)p FG(jj)p FF(d)1911 675 y FB(+)1969 716 y FF(\013)q FG(jj)2088 736 y FH(L)2136 703 y Fx(p)2136 759 y(k)2205 716 y FM(=)f FF(c)p FG(jj)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))k FG(\000)i FF(F)2899 675 y FB(+)2903 744 y(^)2885 761 y FH(A)2938 770 y Fy(0)2976 716 y FG(jj)3032 736 y FH(L)3080 703 y Fx(p)3080 759 y(k)3150 716 y FG(\024)28 b FF(c)p FG(jj)p FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\))p FG(jj)3704 736 y FH(L)3752 703 y Fx(p)3752 759 y(k)3814 716 y FM(+)22 b FF(c)32 b(:)456 587 y FM(\(3.8\))456 892 y(Here)i(and)h(in)e(the)i(rest)f(of)g(this)g(pro)s(of,)f(all)f (constan)m(ts)k(are)e(denoted)h(b)m(y)g FF(c)p FM(.)48 b(F)-8 b(or)456 1008 y FF(k)30 b FM(=)e(0,)k(the)h(last)f(term)g(is)g (b)s(ounded)i(b)s(ecause)f(of)g(the)g FF(C)2549 972 y FB(0)2620 1008 y FM(b)s(ound)g(on)g FF(\036)p FM(.)605 1125 y(The)46 b(Dirac)e(equation)h FF(D)1590 1083 y FB(+)1587 1152 y FH(A)1649 1125 y FF(\036)k FM(=)g(0)c(is)g(equiv)-5 b(alen)m(t)45 b(to)g FF(D)2778 1083 y FB(+)2775 1152 y FH(A)2828 1161 y Fy(0)2867 1125 y FF(\036)k FM(=)g FG(\000)p FF(\015)5 b FM(\()p FF(a)p FM(\))p FF(\036)p FM(,)456 1241 y(where)41 b FF(a)g FM(=)f FF(A)27 b FG(\000)h FF(A)1231 1256 y FB(0)1311 1241 y FM(=)40 b FF(h)28 b FM(+)f FF(\013)q FM(.)65 b(As)41 b FF(h)f FM(is)g(b)s(ounded)h(in)e FF(C)2727 1205 y FC(1)2802 1241 y FM(,)j(b)s(ounding)d(the)456 1357 y(deriv)-5 b(ativ)m(es)32 b(of)g FF(a)h FM(is)f(the)h(same)g(as)f (b)s(ounding)g(those)h(of)f FF(\013)q FM(.)605 1473 y(The)i(basic)e (elliptic)e(estimate)h(for)i(the)g(Dirac)e(op)s(erator)h FF(D)2819 1432 y FB(+)2816 1500 y FH(A)2869 1509 y Fy(0)2940 1473 y FM(giv)m(es:)692 1644 y FG(jj)p FF(\036)p FG(jj)862 1665 y FH(L)910 1642 y Fy(2)910 1686 y(2)975 1644 y FG(\024)c FF(c)p FM(\()p FG(jj)p FF(D)1300 1603 y FB(+)1297 1671 y FH(A)1350 1680 y Fy(0)1388 1644 y FF(\036)p FG(jj)1502 1665 y FH(L)1550 1642 y Fy(2)1550 1686 y(1)1610 1644 y FM(+)22 b FG(jj)p FF(\036)p FG(jj)1878 1661 y FH(L)1926 1642 y Fy(2)1963 1644 y FM(\))28 b FG(\024)g FF(c)p FG(jj)p FF(\015)5 b FM(\()p FF(a)p FM(\))p FF(\036)p FG(jj)2529 1665 y FH(L)2577 1642 y Fy(2)2577 1686 y(1)2636 1644 y FM(+)22 b FF(c)975 1798 y FG(\024)28 b FF(c)p FG(jj)p FF(\015)5 b FM(\()p FF(a)p FM(\))p FF(\036)p FG(jj)1475 1815 y FH(L)1523 1796 y Fy(2)1582 1798 y FM(+)22 b FF(c)p FG(jj)p FF(\015)5 b FM(\()p FF(da)p FM(\))p FF(\036)p FG(jj)2126 1815 y FH(L)2174 1796 y Fy(2)2233 1798 y FM(+)22 b FF(c)p FG(jj)p FF(\015)5 b FM(\()p FF(a)p FM(\))p FG(r)2695 1813 y FH(A)2748 1822 y Fy(0)2786 1798 y FF(\036)p FG(jj)2900 1815 y FH(L)2948 1796 y Fy(2)3008 1798 y FM(+)22 b FF(c)33 b(;)456 1959 y FM(where)h(again)d FF(c)i FM(stands)h(for)f(all)e (constan)m(ts.)46 b(The)34 b(\014rst)f(and)g(second)h(terms)f(are)456 2075 y(b)s(ounded)28 b(b)s(ecause)g(of)f(the)h(b)s(ounded)g(Sob)s(olev) f(m)m(ultiplication)c FF(L)2911 2039 y FB(4)2962 2075 y FG(\002)11 b FF(L)3116 2039 y FB(4)3184 2075 y FG(!)27 b FF(L)3377 2039 y FB(2)3417 2075 y FM(,)456 2192 y(using)33 b(\(3.8\))f(to)g(b)s(ound)h FF(a)g FM(in)f FF(L)1629 2155 y FB(4)1629 2216 y(1)1669 2192 y FM(.)43 b(The)34 b(last)d(term)h(can)h(b)s(e)g(estimated)f(as)1088 2353 y FG(jj)p FF(\015)5 b FM(\()p FF(a)p FM(\))p FG(r)1410 2368 y FH(A)1463 2377 y Fy(0)1501 2353 y FF(\036)p FG(jj)1615 2369 y FH(L)1663 2350 y Fy(2)1729 2353 y FG(\024)28 b FF(c)p FG(jj)p FF(a)p FG(jj)2039 2372 y FH(L)2087 2340 y Fx(p)2087 2394 y Fy(1)2126 2353 y FG(jjr)2265 2368 y FH(A)2322 2353 y FF(\036)21 b FG(\000)i FF(a\036)p FG(jj)2666 2369 y FH(L)2714 2350 y Fy(2)2785 2353 y FF(;)456 2516 y FM(for)k(some)g FF(p)h(>)g FM(4,)g(where)h(the)f(\014rst)g(term) g(on)f(the)h(righ)m(t{hand{side)f(is)h(b)s(ounded)456 2632 y(b)s(ecause)39 b(of)44 b(\(3.8\),)39 b(and)f(the)g(second)h(is)e (b)s(ounded)i(b)s(ecause)g FG(r)2856 2647 y FH(A)2913 2632 y FF(\036)f FM(and)g FF(\015)5 b FM(\()p FF(a)p FM(\))p FF(\036)456 2748 y FM(ha)m(v)m(e)34 b(already)e(b)s(een)h(b)s (ounded)g(in)f FF(L)1833 2712 y FB(2)1873 2748 y FM(.)605 2864 y(By)25 b(the)g(b)s(ounded)g(Sob)s(olev)f(m)m(ultiplication)19 b FF(L)2338 2828 y FB(2)2338 2889 y(2)2384 2864 y FG(\002)5 b FF(L)2532 2828 y FB(2)2532 2889 y(2)2600 2864 y FG(!)27 b FF(L)2793 2828 y FB(3)2793 2889 y(1)2833 2864 y FM(,)f(the)f FF(L)3112 2828 y FB(2)3112 2889 y(2)3176 2864 y FM(b)s(ound)456 2981 y(on)32 b FF(\036)g FM(giv)m(es)i(an)e FF(L)1122 2945 y FB(3)1122 3005 y(1)1195 2981 y FM(b)s(ound)g(on)h FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\).)43 b(Using)32 b(this)g(in)g(\(3.8\),)g(w)m(e)i(ha)m(v)m(e)g(an)f FF(L)3405 2945 y FB(3)3405 3005 y(2)456 3097 y FM(b)s(ound)38 b(on)g FF(\013)q FM(.)60 b(Using)38 b(this)f(b)s(ound,)j(w)m(e)f(also)f(ha)m (v)m(e)h(an)f FF(L)2651 3061 y FB(3)2651 3122 y(1)2729 3097 y FM(b)s(ound)g(on)g FF(\015)5 b FM(\()p FF(a)p FM(\))p FF(\036)p FM(,)456 3213 y(and)32 b(therefore)h(an)g FF(L)1256 3177 y FB(3)1256 3238 y(2)1328 3213 y FM(b)s(ound)g(on)g FF(\036)p FM(,)f(as)h(in)e(the)i(argumen)m(t)g(ab)s(o)m(v)m(e.)605 3329 y(No)m(w,)f FF(L)919 3293 y FB(3)919 3354 y(2)979 3329 y FG(\002)20 b FF(L)1142 3293 y FB(3)1142 3354 y(2)1210 3329 y FG(!)27 b FF(L)1403 3293 y FB(2)1403 3354 y(2)1474 3329 y FM(is)k(easily)g(b)s(ounded,)h(so)g(that)f(the)h(same)f(argumen) m(ts)456 3446 y(as)h(ab)s(o)m(v)m(e)i(giv)m(e)e FF(L)1118 3409 y FB(2)1118 3470 y(3)1191 3446 y FM(b)s(ounds)h(for)f(b)s(oth)g FF(\013)h FM(and)g FF(\036)p FM(.)605 3562 y(W)-8 b(e)26 b(no)m(w)h(ha)m(v)m(e)g FF(L)1247 3526 y FB(2)1247 3587 y(3)1313 3562 y FM(b)s(ounds)g(on)f FF(a)g FM(and)g(on)g FF(\036)p FM(,)h(whic)m(h)g(w)m(e)g(can)f(impro)m(v)m(e)g(to)f FF(L)3401 3526 y FB(2)3401 3588 y FH(k)456 3678 y FM(b)s(ounds)31 b(for)f(all)f FF(k)34 b FM(b)m(y)d(induction.)42 b(Supp)s(ose)32 b(w)m(e)f(already)g(ha)m(v)m(e)h FF(L)2917 3642 y FB(2)2917 3704 y FH(k)2990 3678 y FM(b)s(ounds)g(for)456 3794 y(some)h FF(k)h FG(\025)c FM(3.)48 b(Then)35 b(the)f(b)s(ounded)h(Sob)s(olev)e (m)m(ultiplication)d FF(L)2900 3758 y FB(2)2900 3820 y FH(k)2966 3794 y FG(\002)24 b FF(L)3133 3758 y FB(2)3133 3820 y FH(k)3206 3794 y FG(!)29 b FF(L)3401 3758 y FB(2)3401 3820 y FH(k)456 3911 y FM(giv)m(es)d(an)g FF(L)883 3874 y FB(2)883 3936 y FH(k)952 3911 y FM(b)s(ound)h(on)f FF(D)1457 3926 y FH(A)1510 3935 y Fy(0)1548 3911 y FF(\036)p FM(,)h(and)g(th)m(us)g(an)f FF(L)2247 3874 y FB(2)2247 3936 y FH(k)r FB(+1)2406 3911 y FM(b)s(ound)g(on)g FF(\036)p FM(.)41 b(In)27 b(the)f(same)456 4033 y(w)m(a)m(y)-8 b(,)34 b(w)m(e)g(ha)m(v)m(e)h(an)e FF(L)1245 3997 y FB(2)1245 4059 y FH(k)1321 4033 y FM(b)s(ound)g(on)g FF(F)1835 3992 y FB(+)1840 4061 y(^)1821 4078 y FH(A)1922 4033 y FM(=)c FF(\033)t FM(\()p FF(\036;)17 b(\036)p FM(\),)32 b(giving)g(an)h FF(L)2874 3997 y FB(2)2874 4059 y FH(k)r FB(+1)3040 4033 y FM(b)s(ound)g(on)456 4161 y FF(a)f FM(via)g(\(3.8\).)605 4277 y(Th)m(us,)37 b(the)e(sequence)j FF(u)1515 4292 y FH(i)1542 4277 y FM(\()p FF(A)1653 4292 y FH(i)1682 4277 y FF(;)17 b(\036)1784 4292 y FH(i)1811 4277 y FM(\))35 b(m)m(ust)g(ha)m(v)m(e)h(a)f(subsequence)j(con)m(v)m (erging)456 4393 y(with)h(all)f(its)h(deriv)-5 b(ativ)m(es)39 b(to)h(a)f(smo)s(oth)g(limit,)f(whic)m(h)i(is)f(also)g(a)h(solution)e (to)456 4509 y(the)33 b(monop)s(ole)d(equations.)p 3374 4509 4 66 v 3378 4447 59 4 v 3378 4509 V 3436 4509 4 66 v 605 4680 a(As)j(a)f(corollary)f(of)h(this)h(pro)s(of)e(w)m(e)j(ha) m(v)m(e:)605 4857 y FL(Cor)n(ollar)-7 b(y)39 b(3.7.)49 b FK(The)36 b(mo)-5 b(duli)36 b(sp)-5 b(ac)g(es)36 b(of)g FF(L)2403 4821 y FB(2)2403 4883 y FH(k)2483 4857 y FK(solutions)g(to)h (the)f(mono-)456 4973 y(p)-5 b(ole)39 b(e)-5 b(quations)48 b FM(\(2.1\))40 b FK(and)49 b FM(\(2.3\))40 b FK(mo)-5 b(dulo)40 b FF(L)2196 4937 y FB(2)2196 4999 y FH(k)r FB(+1)2369 4973 y FK(gauge)g(e)-5 b(quivalenc)g(e)38 b(for)i(the)456 5090 y(di\013er)-5 b(ent)37 b FF(k)g FG(\025)d FM(4)k FK(ar)-5 b(e)38 b(al)5 b(l)37 b(di\013e)-5 b(omorphic)37 b(to)h(e)-5 b(ach)37 b(other,)i(and)f(to)g(the)g(mo)-5 b(duli)456 5206 y(sp)g(ac)g(e)34 b(of)g(smo)-5 b(oth)34 b(solutions.)p eop %%Page: 45 7 45 44 bop 1128 235 a FE(2.)35 b(COMP)-6 b(A)n(CTNESS)33 b(OF)h(THE)f(MODULI)g(SP)-6 b(A)n(CE)570 b(45)605 435 y FD(2.1.)56 b(Tw)m(o)37 b(generalisations.)48 b FM(An)32 b(elab)s(oration)d(of)j(the)g(pro)s(of)f(of)g(Theo-)456 551 y(rem)h(3.6)g(yields)g(the)h(follo)m(wing)d(stronger)j(result:)605 725 y FL(Theorem)38 b(3.8.)49 b FK(The)26 b(functional)h FF(E)18 b FM(:)33 b FG(B)e(!)c FA(R)39 b FK(de\014ne)-5 b(d)26 b(in)i(De\014nition)e(2.1)456 842 y(satis\014es)34 b(the)h(Palais{Smale)e(c)-5 b(ondition.)605 1016 y FM(Giv)m(en)44 b(the)g(b)s(ounds)g(on)g(solutions)f(of)g(the)h(p)s(erturb)s(ed)g (monop)s(ole)e(equa-)456 1132 y(tions)37 b(\(2.4\))h(and)g(\(2.5\))f (pro)m(v)m(ed)j(in)d(Prop)s(ositions)g(3.1)h(and)g(3.2,)h(the)g(pro)s (of)e(of)456 1248 y(the)k(compactness)i(theorem)d(3.6)h(go)s(es)g (through)h(for)e(the)i(mo)s(duli)c(space)43 b FF(M)3394 1263 y FH(!)456 1365 y FM(for)d(an)m(y)i FF(!)t FM(.)69 b(Ho)m(w)m(ev)m(er,)45 b(ev)m(en)e(after)e(gauge)g(transformations,)h (the)f(solutions)456 1481 y(represen)m(ting)33 b(p)s(oin)m(ts)f(in)g FF(M)1507 1496 y FH(!)1591 1481 y FM(are)g(only)g(as)h(regular)f(as)g FF(!)t FM(.)456 1670 y Fq(Notes:)456 1816 y Fp(1.)46 b(Witten)32 b([)p Fo(37)q Fp(])g(w)m(as)h(the)f(\014rst)f(to)i(exploit) e(the)i(in)m(terpla)m(y)e(b)s(et)m(w)m(een)i(the)f(monop)s(ole)456 1924 y(equations)e(and)h(the)g(W)-8 b(eitzen)m(b\177)-45 b(oc)m(k)34 b(form)m(ula,)c(and)h(Kronheimer)e(and)h(Mro)m(wk)-5 b(a)32 b([)p Fo(18)q Fp(])456 2032 y(\014rst)26 b(indicated)g(ho)m(w)h (to)g(deriv)m(e)g(the)g(compactness)h(of)f(the)h(mo)s(duli)c(space)k (from)e(this.)456 2140 y(2.)41 b(A)30 b(pro)s(of)g(of)g(Theorem)g(3.8)i (w)m(as)f(published)26 b(in)j([)p Fo(13)q Fp(].)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF