arXiv:1901.04539v1 [math.DG] 14 Jan 2019 etrbundle vector srfre oa netmt ftenme fbudsae for speci states problems, phys bound nonlinear mathematical metrics. of and for Einstein geometry number estimates both the state of of view index/bound estimate of point an the as From to referred is rtclpit for points Critical hc lasminimize always which aosaypoi oml fWy Wy1 ntegot fei of growth the on [Wey11] Weyl Schr¨odinger operator of a formula asymptotic famous 0 il oncin oeitee nteciia dimension critical the in expects even one exist generically that to known connections also Mills is it [Tau82]), (eg. htfracneto ohv ihidxi utas aehigh have 1.1. also type. Theorem must this it of index estimate C high an the have yields from to result cue connection a a Taking for curvature. that u connection’s the bundle of curvature the the as by Schr¨odinger determined oper a term operator, inhomogeneous Jacobi with relevant the of particul that in is connections, im Yang–Mills becomes non-minimizing it functional these Yang–Mills the connec understand Yang–Mills to connectio Thus, stable Yang–Mills non-minimizing every so [BL81], 4 instanton dimension in Furthermore, ainPsdcoa eerhFlosi.J tet ssup is Streets J. Fellowship. Research Postdoctoral dation h eaiepr of part negative the Suppose . e ( Let h lsia wklLe–oebu CR siae[Cwi7 estimate (CLR) Cwikel–Lieb–Rozenblum classical The ..Grk sspotdb S rn M-813.CL Kell C.L. DMS-1811034. grant NSF by supported is Gursky M.J. Date NE-NRYETMTSFRYN–IL ONCIN AND CONNECTIONS YANG–MILLS FOR ESTIMATES INDEX-ENERGY aur 4h 2019. 14th, January : siaefrEnti erc ntrso h oooyadth and num topology Betti the the curvature. for scalar of estimates rat positive terms invariant growth conformally in derive sharp i metrics we the the Einstein for captures for bound and estimate a invariant, class obtain conformally the we to is connections related Yang–Mills to four-manifolds, over Applied bundles vector on Abstract. X n g , ∇ easot,cmatReana aiod n suppose and manifold, Riemannian compact smooth, a be ) ATE .GRK,CSYLN ELHR N EFE STREETS JEFFREY AND KELLEHER, LYNN CASEY GURSKY, J. MATTHEW sannisatnYn–il oncino etrbundle vector a on connection Yang–Mills non-instanton a is E Let over epoeacnomlyivratetmt o h ne fSc of index the for estimate invariant conformally a prove We YM ( X V X 4 YM hscnrlrsl a plctost ahmtclphys mathematical to applications has result central This . r aldYn–il oncin,icuigteseilc special the including connections, Yang–Mills called are g , h agMlseeg soitdto associated energy Yang–Mills The . L ) = eacoe,oine ormnfl,wt aaeinvariant Yamabe with four-manifold, oriented closed, a be hnte xs.Wieteeaemn xsec eut o i for results existence many are there While exist. they when − + ∆ YM ISENMETRICS EINSTEIN V nabudddmi in domain bounded a on 1. [ ∇ Introduction := ] Z X 1 otdb S rn DMS-1454854. grant NSF by ported n | F ∇ | 2 tratn nLeagbavle 1-forms, algebra-valued Lie on acting ator rt nesadteridx hsindex This index. their understand to ar dV si hsstigwl aepstv index. positive have will setting this in ns clCie–ibRznlmestimate. Cwikel–Lieb–Rozenblum ical eso noine ormnfl with four-manifold oriented an of bers hri upre yaNtoa cec Foun- Science National a by supported is eher clyfrYn–il oncin and connections Yang–Mills for fically .Frhroew eiea index an derive we Furthermore e. n o-ntno,nnmnmzn Yang– non-minimizing non-instanton, otn oudrtn h tutr of structure the understand to portant g isenHleteeg.Lastly energy. Einstein–Hilbert e dxi em fiseeg which energy its of terms in ndex . dryn imninmti swell as metric Riemannian nderlying SU9 M0 S2 Bor92]. SS92, HM90, [SJU89, 4 = inwt ml ag ru san is group gauge small with tion Retmt n a oeroughly hope may one estimate LR ∇ agMlseeg.Tefis main first The energy. Yang–Mills c,i sipratt n similar find to important is it ics, ,Le6 o7] eae othe to related Ros72], Lie76, 7, R h ierSchr¨odinger operator. linear the sgvnby given is evle,bud h ne of index the bounds genvalues, n ro igroeaosacting hr¨odinger operators ntrso the of terms in E ∇ over sacneto na on connection a is aso instantons, of lass X 4 ihstructure with c,weeit where ics, Y( L n 2 X nstantons omof norm 4 , [ g ]) > 2 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS group G SO(E), and curvature F . Let ı( ) denote the index and ν( ) the nullity of . Then ⊂ ∇ ∇ ∇ ∇ 2 144e dim(gE) 2 4 2 ı( )+ ν( ) 4 2 12π χ X + 12 F dVg ∇ ∇ ≤ Y(X , [g]) − X | ∇| n Z
2 + 3√2 Wg F dVg +3 Wg dVg , X | || ∇| X | | Z Z o 4 where e is Euler’s number, χ(X ) is the Euler characteristic and Wg is the Weyl tensor.

If is an instanton, then ν( ) = 0 and the Atiyah–Singer index formula gives an explicit formula for ı(∇ ) depending on topological∇ data (see Chapter 4 of [DK90]). Our statement explicitly does not include∇ this case, and we use the assumption of nonvanishing of F + when constructing a metric conformal to the base, with respect to which we carry out the index estimate∇ (see Proposition 3.6). When the base manifold is the round sphere we can simplify the statement to the following:

4 Corollary 1.2. Let E (S , gS4 ) be a vector bundle over the round sphere with structure group G SO(E), with a non-instanton→ Yang–Mills connection. Then ⊂ ∇

2 1 2 ı( )+ ν( ) 9e dim(gE) 1+ 4π2 F dVg . ∇ ∇ ≤ − S4 | ∇| n Z o An index plus nullity estimate for Yang–Mills connections appeared in [Ura86], under the much stronger assumption that the base manifold has positive Ricci curvature and with a bound depend- ing on the L∞-norm of the bundle curvature. Our result only assumes positive Yamabe invariant, and the bound depends on conformal invariants of the base manifold and the Yang–Mills energy. This is more natural, in view of the fact that the index and nullity are conformal invariants. Fur- thermore, although the constants in Theorem 1.1 are almost certainly not sharp (in fact, the sharp value is not known in the classical CLR inequality; cf. [HKRV18]), we can show by means of exam- ples that the growth rate of the index as a function of the Yang–Mills energy is sharp. Specifically, combining an index estimate of Taubes [Tau83] as well as an explicit construction of non-instanton Yang–Mills connections due to Sadun–Segert [SS92], we exhibit a family of connections whose in- dex grows linearly in the Yang–Mills energy (Proposition 3.9 below). Lastly we point out that the estimate we give in 2 can be adapted to give an index estimate for Yang–Mills connections in any § n dimension in terms of the L 2 norms of F and the Ricci curvature, and the Sobolev constant, and in this case the proof is a very direct adaptation of the method of Li–Yau [LY83] (see Remark 2.6). Our second main result is an index estimate for Einstein metrics in dimension four. Einstein metrics arise as critical points of the normalized total scalar curvature functional

1/2 (1.1) S [g] = Vol(g)− Rg dVg . 4 ZX It is well-known that Einstein metrics are never stable critical points, since S is minimized over conformal variations but is locally maximized over transverse-traceless variations, possibly up to a finite dimensional subspace. The index ı(g) of an Einstein metric is dimension of the maximal subspace on which the second variation is negative when restricted to transverse traceless variations, while the nullity ν(g) is the dimension of the space of infinitesimal Einstein deformations. While there are some works characterizing the stability and space of deformations of Einstein metrics ([Koi79, Koi82, DWW05, DWW07]), it seems very little is known about the index in the case it is positive. Intuitively, one might expect an Einstein metric with large index to have small energy. We derive an estimate of this kind which relies on explicit universal constants and the Euler characteristic. INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEINMETRICS 3

Theorem 1.3. Let (X4, g) be an Einstein four-manifold with positive scalar curvature. Then

χ(X4) S [g] 24π , ≤ s3+ δ [ı(g)+ ν(g)] 1 where δ = 24e2 , and e is Euler’s number. Our final application is a bound on the Betti numbers of an oriented four-manifold X4 of positive scalar curvature. Bounds for the Betti numbers in terms of the curvature, Sobolev constant, and diameter of the manifold were proved by P. Li in [Li80]. These estimates can be viewed as refined or quantitative versions of the classical vanishing theorems; see [B88]´ for a beautiful survey. To state our results we need to introduce two conformal invariants of four-manifolds with positive Yamabe invariant. To define the first conformal invariant, we need some additional notation. Let A = Ag denote the Schouten tensor of g: A = 1 Ric 1 Rg , 2 − 6 where Ric is the Ricci tensor and R the scalar curvature
of g. Let σ2(A) denote the second symmetric function of the eigenvalues of A (viewed as a symmetric bilinear form on the tangent space at each point). Then σ (A)= 1 Ric 2 + 1 R2. 2 − 8 | | 24 The integral of this expression is a scalar conformal invariant of a four-manifold. Using this we define the following two conformal invariants:

4 4 X σ2(A) dV ρ1(X , [g]) := 4 2 , RY(X , [g]) (1.2) + 2 4 24 X W dV ρ+(X , [g]) := | 4 | 2 . Y(R X , [g]) 4 4 + 4 Let b1(X ) denote the first Betti number of X , and let b (X ) denote the maximal dimension of a subspace of Λ2(X4) on which the intersection form is positive. It follows from ([Gur98] Theorem 4 3 2) that if b1(X ) > 0 then ρ1 0, with equality only when conformal to a quotient of S R with ≤ + × the product metric. Furthermore, it follows from ([Gur00] Theorem 3.3) that if b > 0 then ρ+ 1, with equality only when conformal to a K¨ahler metric with positive scalar curvature. Using≥ the general index estimate of Section 2, we can prove quantitative versions of these estimates: Theorem 1.4. Let (X4, g) be an oriented four-manifold with Y(X4, [g]) > 0. Then (1.3) b (X4) 9e2 (1 24ρ ) , 1 ≤ − 1 and (1.4) b+(X4) 3e2 (2√ρ 1)2 , ≤ + − where e is Euler’s number. Here, as in the Yang-Mills estimate, our constants are likely not sharp but the growth rate is. In particular, by taking connect sums with sufficiently long necks, we can produce locally conformally flat metrics on the manifold k#S3 S1 whose Yamabe invariant is uniformly bounded below. × Evidently this manifold has b1 = k, while for these conformal classes we see that the right hand side of (1.3) grows linearly in k. The proofs of these theorems all rely on an extension of the CLR estimate to elliptic operators on vector bundles with certain geometric backgrounds (see Section 2). The case of dimension n = 4 especially requires careful analysis of the curvature terms in the relevant index operator in order to capture the conformal invariance. While many proofs of the classical CLR inequality by now exist, 4 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS the proof of Li–Yau [LY83] gives explicit bounds in terms of the Sobolev constant. By adapting their ideas to operators modeled on the conformal Laplacian but acting on sections of a vector bundle, we are able to obtain estimates in terms of conformal invariants. An important technical step is to compare the L2-trace of the heat kernel of a Schr¨odinger-type operator acting on sections of a vector bundle to the heat trace of an associated scalar operator. Again, many results of this kind exist (see [HSU77, HSU80, Sim79]), but we adapt a proof of Donnely–Li [DL82] as it is closest in spirit to the other estimates. Combining these ideas together with a conformal gauge-fixing argument yields our main index estimates.

Acknowledgements. The authors thank Elliott Lieb, Francesco Lin, Zhiqin Lu, and Richard Schoen for informative discussions.

2. General index estimate In this section we adapt the proof of the Cwikel–Lieb–Rosenblum inequality due to Li–Yau [LY83] to prove an index estimate for a certain class of elliptic operators acting on sections of vector bundles. Given a vector bundle (X4, g) with a metric-compatible connection , let E → 0∇ 4 ∆=∆g : Γ( ) Γ( ) denote the rough Laplacian. Given a non-negative function V C (X ), consider theE operator→ E ∈ (2.1) = ∆+ 1 R V, S − 6 − where R = Rg is the scalar curvature of g. We will assume throughout this section that R 0, and the Yamabe invariant Y(X4, [g]) > 0. Our main result is ≥ Theorem 2.1. If N ( ) denotes the number of non-positive eigenvalues of , then 0 S S 2 2 36e rank( ) V L2 (2.2) N0( ) E k k . S ≤ Y (X4, [g])2 The proof is a consequence of a series of technical lemmas, and will appear at the end of the section. We begin with some notation. We need to distinguish between the Laplacian on functions and the rough Laplacian acting on sections of , so from now on we set E 4 4 ∆ : C∞ X C∞ X , 0 → ∆ : C∞ ( ) C∞ ( ) . E →
E
Fix some small ǫ> 0 define

(2.3) Vǫ := V + ǫ. Consider the two operators 1 1 0 := ∆0 R , P Vǫ − 6
:= 1 ∆ 1 R . P Vǫ − 6 As a first step we give the following analogue of an estimate
in Li–Yau:

0 0 Lemma 2.2. Let µ1 µ2 denote the eigenvalues of 0, counted with multiplicity. Then for all t> 0, ≤ ≤ ··· −P 2 ∞ 2µ0t 36 Vǫ L2 2 (2.4) e− i || || t− . ≤ Y(X4, [g])2 Xi=1 INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEINMETRICS 5

2 Proof. As in [LY83], we take ψi to be an orthonormal basis of L (Vǫ dV) consisting of eigenfunc- tions of : { } −P0 ψ = µ ψ , −P0 i i i with

ψ (x)ψ (x)V (x) dV δ . i j ǫ x ≡ ij ZX Let

tµi H0(x,y,t) := e− ψi(x)ψi(y). Xi=1 Note that H0 is the heat kernel associated to the operator 0 with respect to the weighted inner 2 P product L (Vǫ dV). In particular, ∂ [H (x,y,t)] = H (x,y,t) ∂t 0 P0 0 (2.5) 1 1 = ∆0 R H0(x,y,t). Vǫ − 6 Moreover, since R 0 we have
≥ H0(x,y,t) > 0, and for any f C0 X4 , ∈
(2.6) lim H0(x,y,t)f(y)Vǫ(y) dVy = f(x). t 0 → ZX We also let

2 h(t) := H0(x,y,t) Vǫ(x)Vǫ(y) dVx dVy ZX ZX ∞ 2µ0t = e− i . Xi=1 We now argue as in the proof of Theorem 2 of [LY83]: differentiating h, using (2.5) and integrating by parts, we have

dh = 2 V (x) H (x,y,t)( ) H (x,y,t)V (y) dV dV dt ǫ 0 P0 y 0 ǫ y x ZX ZX = 2 V (x) H (x,y,t) ∆ 1 R H (x,y,t) dV dV . ǫ 0 0 − 6 y 0 y x (2.7) ZX ZX 2
1 2 = 2 Vǫ(x) yH0(x,y,t) + 6 R(y)H0(x,y,t) dVy dVx, − X X |∇ | Z Z h i 2 1 2 = 2 Vǫ(x) yH0(x,y,t) + 6 R(y)H0(x,y,t) dVy dVx . − X X |∇ | Z Z h i By the definition of the Yamabe invariant, 1/2 4 4 2 1 2 Y(X , [g]) H0(x,y,t) dVy 6 yH0(x,y,t) + 6 RyH0(x,y,t) dVy . X ≤ X |∇ |  Z  Z Using this, we can rewrite (2.7) as   1/2 (2.8) dh 1 Y(X4, [g]) V (x) H (x,y,t)4 dV dV . dt ≤− 3 ǫ 0 y x ZX ZX  6 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

To obtain a differential inequality for h we need a further a priori upper bound. Iterating H¨older’s inequality twice and using the fact that H0(x,y,t) > 0 we note (2.9) 2 h(t)= Vǫ(x) H0(x,y,t) Vǫ(y) dVy dVx ZX ZX 1/3 2/3 V (x) H (x,y,t)4 dV H (x,y,t)V 3/2(y) dV dV ≤ ǫ 0 y 0 ǫ y x ZX "ZX  ZX  # 1/2 2/3 2 1/3 V (x) H (x,y,t)4 dV dV V (x) H (x,y,t)V 3/2(y) dV dV , ≤ ǫ 0 y x ǫ 0 ǫ y x "ZX ZX  # "ZX Z  # It remains to estimate the second term on the right hand side above, which is done by treating it as an auxiliary solution to the heat equation. In particular set

3/2 (2.10) Q(x,t) := H0(x,y,t)Vǫ(y) dVy . ZX Note that Q is a solution of the heat equation associated to : P0 ∂ 1 1 ∂t [Q(x,t)] = ( 0Q)(x,t)= V (x) ∆0 6 R Q(x,t), (2.11) P ǫ − 1/2 Q(x, 0) = Vǫ (x).

Note in particular the power of Vǫ, which is a consequence of the weighted inner product. We first compute

d 2 ∂ dt Q(x,t) Vǫ(x) dVx = 2 Q(x,t) ∂t [Q(x,t)] Vǫ(x) dV ZX  ZX = 2 Q(x,t) ∆ 1 R Q(x,t) dV (2.12) 0 − 6 ZX
= 2 Q(x,t) 2 + 1 R(x)Q(x,t)2 dV − |∇ | 6 ZX 0. h i ≤ Integrating this and applying (2.11),

Q(x,t)2V (x) dV Q(x, 0)2V (x) dV ǫ ≤ ǫ ZX ZX 2 = Vǫ(x) dV . ZX Now, using (2.10), 2 2 3/2 Q(x,t) Vǫ(x) dV= Vǫ(x) H0(x,y,t)Vǫ(y) dVy dVx, ZX ZX ZX  and so substituting into (2.12) we obtain

2 1/2 3/2 V 2 V (x) H (x,y,t)V (y) dV dV . k ǫkL ≥ ǫ 0 ǫ y x "ZX ZX  # Substituting this into (2.9), we have

1/2 2/3 h(t) V (x) H (x,y,t)4 dV dV V 2/3. ≤ ǫ 0 y x k ǫkL2 "ZX ZX  # INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEINMETRICS 7

By (2.8), we conclude 4 dh 1 Y(X , [g]) 3/2 (2.13) 3 h(t) . dt ≤− V 2 || ǫ||L Integrating and using the fact that h(t) as t 0+ we conclude →∞ → 2 36 Vǫ L2 2 h(t) || || t− , ≤ Y(X4, [g])2 which is equivalent to (2.4).  The key lemma that allows us to pass from Lemma 2.2 to Theorem 2.1 is the following: Lemma 2.3. We have

t t 0 (2.14) tr 2 e P rank( ) tr 2 e P . L ≤ E L Proof. This is based on argument in [DL82], Theorem 4.3 and Corollary 4.4. Let H(x,y,t) denote the heat kernel associated to with respect to the weighted inner product of Lemma 2.2. More P precisely, let µ1 µ2 denote the eigenvalues of , counted with multiplicity, and let φi be an orthonormal≤ basis≤··· of sections of L2( ,V dV) consisting−P of eigenfunctions of : { } E ǫ −P φ = µ φ , −P i i i with φ (x), φ (x) V (x) dV = δ . h i j i ǫ x ij ZX Then the associated heat kernel is given by tµ H (x,y,t)= e− i φ (x) φ (y) . i ⊗ i Xi=1 If H denotes the norm of H as an endomorphism H( ,x,y) : x y, then H is a subsolution of| (2.5)| (in the sense of distributions): · E → E | | ∂ [ H (x,y,t)] H (x,y,t) ∂t | | ≤ P0 | | (2.15) 1 1 = ∆0 H R H (x,y,t) , Vǫ | |− 6 | | see Lemma 4.1 of [DL82]. Also, in analogy with (2.6), for any
f C0 X4 we have ∈
(2.16) lim H (x,y,t) f (y) Vǫ (y) dVy = f (x) . t 0 | | → ZX By (2.6) and (2.16), (2.17) H (x,y,t) H (x,y,t) | | − 0

= lim H (x,z,t)H0(z,y,τ)Vǫ(z) dVz H (x,z,τ) H0 (z,y,t) Vǫ (z) dVz τ 0 | | − | | → ZX ZX  t = d H (x,z,s) H (z,y,t s) V (z) dV ds ds | | 0 − ǫ z Z0 ZX  t = d [ H (x,z,s)] H (z,y,t s) V (z) dV ds ds | | 0 − ǫ z Z0 ZX  T1 t + H (x,z,s) d [H (z,y,t s)] V (z) dV ds . | | ds 0 − ǫ z Z0 ZX  T2 8 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

We manipulate the second term T2 using (2.5), (2.18) t ∂ T2 = H (x,z,s) ∂s H0 (z,y,t s) Vǫ(z) dVz ds 0 X | | − Z Zt 1 1 = H (x,z,s) ∆0 R H0 (z,y,t s) Vǫ(z) dVz ds − | | Vǫ(z) − 6 − Z0 ZX t
t = H (x,z,s)∆ H (z,y,t s) dV ds+ ∂ H (x,z,s) 1 R (z) H (z,y,t s) dV ds . − | | 0 0 − z ∂s | | 6 0 − z Z0 ZX Z0 ZX Integrating by parts in the term involving ∆0 and using (2.15), reincorporating T2 into (2.17),

t (2.19) H (x,y,t) H (x,y,t)= ∂ H (x,z,s)H (z,y,t s)q (z) dV 0. | | − 0 ∂s − P0 | | 0 − ǫ z ≤ Z0 ZX
Therefore, if tr H denotes the pointwise trace of H( ,x,x) : , g · Ex →Ex t trL2 e P = trgH(x,x,t)Vǫ(x) dVx ZX rank( ) H (x,x,t)V (x) dV ≤ E | | ǫ x ZX rank( ) H (x,x,t)V (x) dV ≤ E 0 ǫ x ZX t 0 = rank( ) tr 2 e P . E L The result follows. 

Combining Lemma 2.2 with Lemma 2.3 we have

Proposition 2.4. Let µ1 µ2 denote the eigenvalues of , counted with multiplicity. Then for all t> 0, ≤ ≤ ··· −P

2 ∞ 2µ t 36 rank( ) Vǫ L2 2 (2.20) e− i E || || t− . ≤ Y(X4, [g])2 Xi=1 Proof. Observe that

∞ 2µit (2t) (2.21) e− = trL2 e P . Xi=1   But by Lemma 2.3,

∞ 0 (2t) (2t) 0 2µ t (2.22) tr 2 e P rank( ) tr 2 e P = rank( ) e− i . L ≤ E L E i=1

X Thus the result follows from Lemma 2.2. 

Corollary 2.5. Let µ denote the kth-eigenvalue of . Then k −P 2 2 36e rank( ) Vǫ 2 (2.23) E k kL µ2 k. Y(X4, [g])2 k ≥ INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEINMETRICS 9

Proof. As in [LY83], take t = 1 in (2.20), then µk 2 36 rank( ) Vǫ 2 ∞ E k kL µ2 exp( 2 µi ) Y(X4, [g])2 k ≥ − µk Xi=1 k exp( 2 µi ) ≥ − µk i=1 X 2 ke− . ≥ The result follows. 

Proof of Theorem 2.1. By the argument of Birman–Schwinger, the number of non-positive eigen- 1 values of the operator ∆+ 6 R + Vǫ is less than or equal to the number of eigenvalues of the 1 − 1 operator = ( ∆+ R) that are less than or equal to 1. But by (2.23), if µk the greatest −P Vǫ − 6 eigenvalue of that is less than or equal to 1, then −P 2 2 36e rank( ) Vǫ 2 k E k kL . ≤ Y(X4, [g])2 Therefore, taking ǫ 0 we conclude → 2 2 36e rank( ) V 2 N ( ) E k kL , 0 S ≤ Y(X4, [g])2 which completes the proof.  Remark 2.6. If (Xn, g) is a vector bundle, n 3, and = ∆+ V is a linear operator acting on sectionsE of →E with V 0, then the preceding≥ argumentsS can− easily be adapted to give an estimate for the number of non-positive≥ eigenvalues of . If C (g) denotes the Sobolev constant, S S n−2 2 n n 2 2 C (g) f n−2 dV f + f dV, S | | ≤ |∇ | ZX  ZX then   rank( ) N ( ) c (1 + V ) n/2 . 0 n En Ln/2 S ≤ CS (g) 2 k k 3. Index estimate for Yang-Mills connections 3.1. Background. Let (E, h) (Xn, g) be a vector bundle with metric over a closed Riemannian → manifold with structure group G SO(E). Let Γ(E) denote the smooth sections of E, and gE denote the associated Lie algebra of⊂ E. For each point x Xn choose a local orthonormal basis of n i ∈ α T X given by ei with dual basis e and a local basis for E given by µα with dual basis (µ∗) { } p { } { } p p { } of the dual E∗. Let Λ denote the space of smooth p-forms over X and set Λ (E) := Λ Γ(E). Given an element in Λp(E) its components are understood be with respect to the forgoing⊗ bases. We will also use the fact that when p = 1, we can take tensor products of the basis elements i α 1 e , µα , (µ∗) to obtain a (local) basis of Λ (E). { We} { will} use{ the} following conventions for the various inner products that appear: 1 η,ω 2 = ηijωij, ν,µ 2 = νijµij, h iΛ 2 h iS0 (X) Xi,j Xi,j 1 1 β α A, B = trE (AB)= A B h igE − 2 − 2 α β Xα,β 1 α β 1 α β P,Q 1 = P Q , R,S 2 = R S . h iΛ (gE ) − 2 iβ iα h iΛ (gE ) − 4 ijβ ijα 10 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

Here, repeated Latin indices indicate contractions by the metric g on Xn, and the components are with respect to the orthonormal basis above. Unless specified otherwise, we will use Einstein summation notation for both bundle and base components. 1 We need certain algebraic actions as well. First there is the bracket operation [, ] : Λ (gE) Λ1(g ) Λ2(g ) defined by × E → E [A, B]β := Aβ Bδ Bβ Aδ , A,B Λ1(g ). jkα jδ kα − kδ jα ∈ E 2 2 1 Also, given η S (T X) and Φ Λ (gE), we may view both as elements of End(Λ (gE)) via the formulas ∈ ∈ β β (η (A))iα = ηijAjα, ([Φ, A])β = [Φ , A ]β =Φβ Aµ Aβ Φµ . iα ji j α jiµ jα − jµ jiα We next recall the definition of the Jacobi operator of . YM Theorem 6.8 of [BL81]. Suppose is a Yang–Mills connection on a vector bundle E over Xn with structure group G SO(E), and ∇ is a one parameter family of connections with . ⊂ {∇s} ∇≡ ∇s|s=0 Furthermore, suppose B := ∂ [ ] Λ1(g ). Then ∂s ∇s s=0 ∈ E 2 d ds2 [ ( s)] = 2 ∇ (B) ,B Λ1(g ) dV, YM ∇ s=0 X J E Z where j j j ∇ (B) = ∆B B + 2 F ,B + Ric B , J i − i −∇i∇ j ji i j where ∆ = a denotes the rough Laplacian.   ∇ ∇a The operator ∇ is degenerate elliptic, due to the action of the infinite dimensional gauge group. Questions of indexJ and nullity always refer to the operator restricted to divergence-free sections B, one which the operator takes the simpler form:

j j (3.1) ∇ (B) = ∆B + 2 F ,B + Ric B . J i − i ji i j The index and nullity of a Yang-Mills connection are underst ood to be those quantities associated to this operator. It follows from the conformal invariance of the Yang-Mills energy that both the index and nullity are conformally invariant.

3.2. Linear algebraic estimates. In this subsection we obtain linear algebraic estimates which enter into estimating the Jacobi operator. The key point is Proposition 3.4, which provides a sharp inequality between the operator and Hilbert-Schmidt norms of the bilinear form appearing in the 2 2 Jacobi operator. Let Z S0 (T X) and Φ Λ (gE); in the following we can view both as elements 1 ∈ ∈ of End(Λ (gE)). Lemma 3.1. Suppose E (Xn, g) is a vector bundle. Then Z and Φ, viewed as endomorphisms 1 → 1 of Λ (gE), are symmetric. Moreover, Z is trace-free as an endomorphism of Λ (gE).

Proof. Take A, B Λ1(g ). Using the symmetry of both Z and the inner product on E, ∈ E 1 β α Z (A) ,B 1 = Zij A B h iΛ (gE ) − 2 jα iβ = 1 Z Bβ Aα − 2 ji iα jβ = Z (B) , A 1 . h iΛ (gE ) INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 11

The symmetry of Z follows. Next, using the cyclicity of inner products over gE, reindexing and skew symmetry of the bracket operation and Φ,

1 β α [Φ, A] ,B 1 = [Φij, Ai] B h iΛ (gE ) − 2 α jβ = 1 Φβ Aδ Bα + 1 Aβ Φδ Bα − 2 ijδ iα jβ 2 iδ ijα jβ = 1 Aδ Bα Φβ + 1 Aβ Φδ Bα − 2 iα jβ ijδ 2 iδ ijα jβ = 1 Aβ Bδ Φα + 1 Aβ Φδ Bα − 2 iδ jα ijβ 2 iδ ijα jβ = 1 Aβ [B , Φ ]δ − 2 iδ j ij β = 1 Aβ [Φ ,B ]δ − 2 iδ ji j β = [Φ,B] , A 1 , h iΛ (gE ) hence Φ is symmetric as an endomorphism. 1 To show that Z is trace-free as an operator on Λ (gE), we construct an orthonormal basis for 1 Λ (gE) as described at the beginning of Section 3.1: for fixed (k, α, β), let

k α (3.2) A := e (µ∗) µ , (k,α,β) ⊗ ⊗ β where ei is a basis of T M that diagonalizes Z. Note that the components of these basis elements are given{ } by

A ν = δ δν δβ, α = β, (k,α,β) ℓµ kℓ α µ 6
so the only nonzero entry is the (k, α, β)-component. Computing the trace of Z with respect to this basis yields

1 ν µ Z(A(k,α,β)), A(k,α,β) 1 = Zij A(k,α,β) A(k,α,β) Λ (gE ) − 2 iµ jν = 1 Z δ δν δβδ
δµδβ
− 2 ij ki α µ kj α ν = 1 Z δβδβ − 2 ii α α = 0, since Z is traceless on T M. The result follows. 

1 Lemma 3.2. As operators on Λ (gE), the ranges of Z and Φ are orthogonal subspaces.

Proof. The orthogonality of Z and [Φ, ] will follow since Z preserves the bundle components while Φ is skew symmetric with respect to the· bundle components. Using the basis (3.2) as above, for fixed (k, α, β), then

ν ν Z A(k,α,β) iµ =Zℓi A(k,α,β) ℓµ ν β

=Zℓi δkℓδαδµ
ν β =Zki δαδµ Z if µ = α, β = ν, = ki . (0 otherwise. 12 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

Similarly, Φ, A ν =Φν A δ A ν Φδ (k,α,β) iµ ℓiδ (k,α,β) ℓµ − (k,α,β) ℓδ ℓiµ   =Φν δ δδ δβ
δ δνδβΦδ
ℓiδ kℓ α µ − kℓ α δ ℓiµ =Φν δβ δν Φβ kiα µ − α kiµ 0 α = ν and β = µ Φβ α = ν and β = µ = − kiµ 6 . Φν α = ν and β = µ  kiα 6 0 α = ν and β = µ  6 6  2 Where here, we are noting that since Φ Λ (gE), its endomorphism indices cannot coincide.  ∈ To state our next result, we need to introduce an algebraic invariant defined by Bourguignon– Lawson. Let [A,B] γ0 := sup | A B | . A,B Γ(g ) 0 | || | ∈ E \{ } Lemma 2.30 of [BL81] gives the universal upper bound (3.3) γ √2, 0 ≤ and characterizes the case of equality. Lemma 3.3. If A Λ1(g ), then ∈ E n 1 2 (3.4) [A, A] 2 γ0 − A 1 , | |Λ (gE ) ≤ 2n | |Λ (gE ) q Since γ √2, in general we have 0 ≤ n 1 2 (3.5) [A, A] 2 − A 1 . | |Λ (gE ) ≤ n | |Λ (gE ) q Proof. Fix a point p Xn and let ei to be an orthonormal basis of Λ1. If A Λ1(g ), then we ∈ ∈ E can express A = A ei for A Γ (g ). Then i i ∈ E 2 1 β α [A, A] 2 = [A, A] [A, A] | |Λ (gE ) − 4 ijα ijβ 2 1 = 2 [A, A]ij gE Xi,j 1 2 = [Ai, Aj] 2 | |gE Xi,j 2 = [Ai, Aj] . | |gE Xi

2 2 [A, A] 2 = [Ai, Aj] | |Λ (gE )  | |gE  Xi

Now

4 2 2 2 2 4 A 1 = Ai Aj = 2 Ai Aj + Ai , | |Λ (gE ) | |gE | |gE | |gE | |gE | |gE Xi,j Xi

2 4 1 2 1 4 Ai g n Ai g = n A Λ1(g ). | | E ≥ | | E ! | | E Xi Xi Therefore,

2 2 (n 1) 4 Ai Aj − A 1 . | |gE | |gE ≤ 2n | |Λ (gE ) 1 i

2 2 n 1 4 [A, A] 2 γ − A 1 , | |Λ (gE ) ≤ 0 2n | |Λ (gE ) and taking the square root yields (3.4).
 Proposition 3.4. Suppose E (Xn, g) is a vector bundle and let → = Z+[Φ, ] : Λ1(g ) Λ1(g ). B · E → E Then

n 1 2 2 2 2 (A, A) −n Z 2 ∗ + 2γ0 Φ 2 g A Λ1(g ). |B |≤ · | |S0 (T M) | |Λ ( E ) | | E q q  1 Proof. Since is symmetric by Lemma 3.1, there exists an orthonormal basis of Λ (gE) with respect to which theB matrix of is diagonalized. Since the ranges of Z and Φ are orthogonal by Lemma 3.2, we can express theB matrix of as B ~z 0 [ ]= , B 0 φ~   where ~z 0 0 0 [Z] = , [Φ] = , 0 0 0 φ~     are the matrices of Z and Φ with respect to this basis, ~z = (z1, , zn), φ~ = (φ1, , φN ) are the eigenvalues of Z and Φ respectively. If A Λ1(g ), then we can· write · · A = A + A· ·, · where ∈ E 1 2 ~a 0 A = , A = , 1 0 2 ~b     with ~a = (a , , a ), ~b = (b , , b ). Therefore, as a bilinear form 1 · · · n 1 · · · N (A, A)=Z(A , A )+Φ(A , A ) B 1 1 2 2 ~z 0 ~a = ~a ~b 0 φ~ ~b ·      2 2 = ziai + φjbj . Xi Xj 14 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

Since Z is trace-free via Lemma 3.1,

2 Z(A1, A1) = ziai | | Xi n 1 2 − ~z ~a ≤ n | || | q n 1 2 = − Z 2 ∗ A1 1 . n | |S0 (T M) | |Λ (gE ) Also, by Lemma 3.3, q

Φ(A , A ) = φ b2 | 2 2 | j j j X = [Φ, A ] , A 1 2 2 Λ (gE ) |h i| = 2 Φ, [A2, A2] 2 |h i|Λ (gE ) 2 Φ 2 [A2, A2] 2 ≤ | |Λ (gE ) | |Λ (gE ) n 1 2 2γ0 Φ 2 − A2 1 . ≤ | |Λ (gE ) 2n | |Λ (gE ) Therefore, q

n 1 2 2 (A, A) − Z 2 ∗ A1 1 + √2γ0 Φ 2 g A2 1 , |B |≤ n | |S0 (T M) | |Λ (gE ) | |Λ ( E ) | |Λ (gE ) q where we have dropped the subscripts designating the norms in order to simplify notation. By the Cauchy-Schwartz inequality,

n 1 2 2 2 4 4 (A, A) −n Z 2 ∗ + 2γ0 Φ 2 g A1 1 g + A2 1 g |B |≤ · | |S0 (T M) | |Λ ( E ) · | |Λ ( E ) | |Λ ( E ) q q n 1 q 2 2 2 2 −n Z 2 ∗ + 2γ0 Φ 2 g A Λ1(g ). ≤ · | |S0 (T M) | |Λ ( E ) | | E q The result follows. q   3.3. A canonical conformal representative. Since the index and nullity of a Yang–Mills con- nection in four dimensions are conformally invariant, we may estimate them with respect to any metric conformal to the base metric g. In this subsection, we specify a choice of conformal metric based on our work in [GKS18]. To this end, suppose is a Yang–Mills connection on a vector bundle E over (X4, g) with structure group G SO(E),∇ and denote the curvature by F = F . For t 0, define ⊂ ∇ ≥ Φt = R t 2√6 W + 3γ F , g g − | |g 1| |g where Rg is the scalar curvature of g, Wg is the Weyl tensor, and γ1(E) is the constant given by ω, [ω,ω] (3.7) γ1(E):= sup h 3 i . ω Λ2 (g ) 0 ω ∈ + E \{ } | |

2 Remark 3.5. The definition of the inner product on Λ+(gE) given in [GKS18] differs from the definition of this paper. In particular, the estimate for γ1(E) in Section 2 of [GKS18] needs to be adjusted. With respect to our current conventions, we have the estimate

2√6 γ1(E) γ0(E) (3.8) ≤ 3 4√3 . ≤ 3 INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 15

We also define the associated operator Lt = 6∆ +Φt . g − g g In [GKS18], based on the ideas of [Gur00], we defined the related curvature and operator Φ = R 2√6 W + 3γ F + , (3.9) g g − | |g − 1| |g L = 6∆ +Φ . g − g g It is easy to see that the expression γ (E) F is independent of the choice of norms. Therefore, 1 | | despite the difference of conventions pointed out in Remark 3.5, the definition of Φg in (3.9) agrees with the corresponding formula (3.5) in [GKS18]. Observe that 0 Φg = Rg, Φ1 Φ . g ≤ g In addition, Φt satisfies the same kind of conformal transformation formula as Φ: giveng ˆ = u2g, t 3 t Φgˆ = u− Lgu. t t If λ1(L ) denotes the first eigenvalue of L , t φL φ dVg (3.10) λ (Lt )= inf X g , 1 g ∞ 2 φ C (X) 0 φ dVg ∈ \{ } R X t then the sign of λ1(L ) is a conformal invariant (see [Gur00],R Proposition 3.2). In particular, by t using an eigenfunction associated with λ1(L ) as a conformal factor, it follows that [g] admits a t t metricg ˆ with Φgˆ > 0 (resp., = 0,< 0) if and only if λ1(Lg) > 0 (resp. = 0,< 0). Proposition 3.6. Assume (X4, [g]) has Y(X4, [g]) > 0. Given a Yang-Mills connection which is not an instanton, there exists t (0, 1] such that λ (Lt0 ) = 0∇. In particular, we can choose a 0 ∈ 1 g conformal metric gˆ [g] with respect to which Φt0 0, hence ∈ gˆ ≡ (3.11) R = 2√6t W + 3γ t F . gˆ 0| gˆ| 1 0| |gˆ Moreover, Y(X4, [g]) (3.12) t0 1. 2√6 W 2 + 3γ F 2 ≤ ≤ k kL 1k kL Proof. Using the Bochner formula for Yang-Mills connections, in [GKS18] we showed that either + 1 F 0, or else λ1(Lg)= λ1(Lg) 0. Since we are ruling out the former by assumption, the latter ≡ ≤ 1 condition must hold. In fact, we can assume λ1(Lg) < 0, since otherwise we could take t0 = 1. t 0 Clearly, λ1(Lg) depends continuously on the parameter t. Since Φg = Rg and the Yamabe 4 0 invariant of (X , [g]) is positive, we know that λ1(Lg) > 0. By the intermediate value theorem, it follows there is t (0, 1] with λ (Lt0 ) = 0. Also, integrating (3.11) and using the Cauchy-Schwarz 0 ∈ 1 g inequality it is easy to see that t0 satisfies (3.12).  3.4. The Proof of Theorem 1.1. In this subection use Theorem 2.1 to give the proof of Theorem 1.1. As remarked above, since the index and nullity are conformal invariants we are free to make a conformal modification of the base metric and we choose the conformal gauge guaranteed by Proposition 3.6. To begin we obtain an algebraic estimate for the Jacobi operator. Specifically, let Z now denote the trace-free Ricci tensor, i.e. Z := Ric 1 Rg. − 4 16 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

We express as J ∇ 1 ∇ = ∆+ 4 R + Z +2[F , ] (3.13) J − ∇ · = ∆+ 1 R + 1 R + √3 γ t [F, ] + Z+ 2 √3 γ t [F, ] , − 6 12 12 1 0 · − 12 1 0 · A B and proceed to estimate the zeroth-ordern operators o andn labeled above. o A B Lemma 3.7. As a bilinear form, 0. A≥ Proof. If we take Z=0 and Φ= F in Proposition 3.4, then ∇ F (A, A) = [F, A], A | | |h i| √3 2γ2 F 2 A 2. ≤ 2 0 | | | | √ q = 6 γ F A 2. 2 0| || | Since γ √2, it follows that 0 ≤ [F, A], A √3 F A 2. |h i| ≤ | || | Therefore, (A, A)= 1 R A 2 + √3 γ t [F, A] , A A 12 | | 12 1 0 h i 1 R A 2 √3 γ t √3 F A 2 ≥ 12 | | − 12 1 0 | | | | = 1 (R 3γ t F ) A 2 .  12 − 1 0 | | | | Using the formula for the scalar curvature in (3.11), we conclude (A, A) 1 R 3γ t F A 2 A ≥ 12 − 1 0 | | | | √  2  = 6 t W A 6 0 | | | | 0. ≥ 

Lemma 3.8. Let (3.14) α = 2 √3 γ t > 0. − 12 1 0 Then 1/2 (3.15) (A, A) 3 Z 2 + 3α2 F 2 A 2 . B ≥− 4 | | | | | | h i Proof. Note that = Z+α[F, ]. If we take Φ = αF in Proposition 3.4 and use the fact that B · γ √2, then 0 ≤ 1/2 (A, A) √3 Z 2 + 2γ2α2 F 2 A 2 B ≥− 2 | | 0 | | | | h 1/2 i 3 Z 2 + 3α2 F 2 A 2 , ≥ 4 | | | | | | as claimed. h i  In view of (3.13) and Lemmas 3.7 and 3.8, we have 1/2 1 3 2 2 2 ∇A, A 2 ∆+ R Z + 3α F A, A 2 hJ iL ≥ h − 6 − 4 | | | | iL 1 = ∆+ R Vh A, A 2 , i
h − 6 − iL
INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 17 where 1/2 (3.16) V = 3 Z 2 + 3α2 F 2 . 4 | | | | We therefore define h i (3.17) = ∆+ 1 R V. S − 6 − To estimate the index and nullity of ∇ it suffices to obtain the estimate for . Applying J 1 S Theorem 2.1 to the operator on the bundle Λ (gE), which has rank 4d, where d = dim(gE), we obtain S 2 144e d 2 N0( ) 4 2 V dV S ≤ Y(X , [g]) X (3.18) Z 144e2d 3 Z 2 dV+3α2 F 2 dV . ≤ Y(X4, [g])2 4 | | | | ( ZX ZX ) By the Chern–Gauss–Bonnet formula

(3.19) 3 Z 2 dV = 12π2χ X4 + 3 W 2 dV+ 1 R2 dV . 4 | | − 2 | | 16 ZX ZX ZX Using the conformal gauge fixing of Proposition
3.6, we can estimate the scalar curvature term above as 2 2 1 2 t0 √ 16 R dV = 16 2 6 W + 3γ1 F dV X X | | | | Z Z   = 3 t2 W 2 dV+ 3√6 γ t2 W F dV+ 9 γ2t2 F 2 dV . 2 0 | | 4 1 0 | || | 16 1 0 | | ZX ZX ZX Substituting this into (3.19) gives

3 Z 2 dV = 12π2χ X4 + 3 1+ t2 W 2 dV 4 | | − 2 0 | | ZX ZX

+ 3√6 γ t2 W F dV+ 9 γ2t2 F 2 dV . 4 1 0 | || | 16 1 0 | | ZX ZX We now substitute this into (3.18) to get 144e2d N ( ) 12π2χ X4 + 3 1+ t2 W + 2 dV 0 S ≤ Y(X4, [g])2 − 2 0 | | (3.20) ZX n

+ 3√6 γ t2 W F dV+ 3α2 + 9 γ2t2 F 2 dV . 4 1 0 | || | 16 1 0 | | ZX ZX
o We estimate the coefficients of each of terms above as follows: For the first coefficient, since t0 1 we have ≤ 3 1+ t2 3. 2 0 ≤ Since 0 t 1 and by (3.8) γ 4√3 , we can bound
the second coefficient by ≤ 0 ≤ 1 ≤ 3 3√6 2 3√6 γ1t γ1 (3.21) 4 0 ≤ 4 3√2. ≤ For the third coefficient we use the formula for α in (3.14) to write (3.22) 3α2 + 9 γ2t2 = 5 (γ t )2 √3(γ t ) + 12. 16 1 0 8 1 0 − 1 0
18 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

Now γ t 4√3 , and the quadratic polynomial q(x) = 5 x2 √3x + 12 attains its maximum at 1 0 ≤ 3 8 − 4√3 x = 0 on the interval 0, 3 . Consequently, h i 3α2 + 9 γ2t2 12. 16 1 0 ≤ With these estimates on the coefficients, we can rewrite
(3.20) as 144e2d 2 4 2 √ 2 N0( ) 4 2 12π χ X + 3 W dV+3 2 W F dV +12 F dV , S ≤ Y(X , [g]) − X | | X | || | X | | n Z Z Z o finishing the proof.
 3.5. Linear growth rate in four dimensions. Theorem 1.1 exhibits that the index can grow at worst linearly in the Yang-Mills energy of the connection. In this section we show that this growth rate is sharp through an explicit family of examples. Various authors [SJU89, HM90, SS92, Bor92] have shown the existence of families of noninstanton Yang–Mills connection for a given SU(2) bundle over S4 provided that the charge κ satisfies κ(E) = 1. We will use the work of Sadun– Segert [SS92], who constructed non-instanton Yang-Mills c6 onnections± on the so-called ‘quadrupole bundles.’ The proposition below analyzes this construction in conjunction with an index estimate of Taubes ([Tau83] Theorem 1.1) to exhibit the required index growth. Proposition 3.9. Given l = 4k 1 > 1, let l denote the Sadun–Segert connection on the quadrupole bundle P S4. There− exists a constant∇ δ > 0 so that (l,3) → l 2 ı δ F l L2 ∇ ≥ || ∇ || Proof. We assume familiarity with the results  and notation of [SS92]. The quadrupole bundles are defined by different lifts of the unique irreducible representation of SU(2) on R5, and are classified by a pair of odd positive integers (n+,n ), with the bundle denotes P(n+,n−). The construction of − [SS92] further restricts to the case n = 1. We will choose n+ = l = 4k 1 > 1, n = 3, and let ± 6 − − l denote the Sadun–Segert connection on P . As computed in [SS89, ASSS89] one has ∇ (l,3) 1 2 2 1 2 (3.23) κ(P(n+,n−))= 8 (n+ n )= 8 l 9 . − − − Furthermore, as the connection l is not self-dual, [Tau83] Theorem
1.1 yields ∇ (3.24) ı( l) 2 κ(P ) + 1 ∇ ≥ (l,3) We claim that there exists a constant C > 0 so that l satisfies
∇ 2 2 (3.25) F l L2 Cl . || ∇ || ≤ Assuming this for the moment, putting together (3.23) - (3.25) yields ı( l) 2 κ(P ) + 1 ∇ ≥ (l,3) 1 2 = 2 l 9 + 1
8 − 1 l2 ≥ 4

1 2 4C F l L2 , ≥ || ∇ || as required. We now prove line (3.25). Connections with quadrupole symmetry on these bundles are described π R in terms of a triple of functions ai : (0, 3 ) , i = 1, 2, 3. The bundle on which the connection is defined is determined by the boundary data.→ In particular, as per ([SS92] Definition 2.5, Lemma 2.6), we require that a = (a1, a2, a3) satisfies (3.26) lim a (θ) = (0, 0, l) , lim a(θ)=(0, 3, 0) , θ 0 θ π → → 3 INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 19 and moreover each a extends to ( ǫ, π + ǫ) such that for all θ ( ǫ,ǫ), i − 3 ∈ − a1 (θ)= a2 ( θ) , a3 (θ)= a3 ( θ) (3.27) − − a π + θ = a ( π θ), a π + θ = a π θ 1 3 3 3 − 2 3 2 3 − We can construct a test connection
which satisfies these cond
itions as follows.
First set a 0. Fix 1 ≡ some small δ > 0 and define a2 via a (θ) 0 for θ ( δ, δ) 2 ≡ ∈ − a (θ) 3 for θ π δ, π + δ 2 ≡ ∈ 3 − 3 0 a (θ) 3 for θ [0, π ] ≤ 2 ≤ ∈ 3
π 0 a′ (θ) 5 for θ [0, ]. ≤ 2 ≤ ∈ 3 and we define a3 via a (θ) 3 for θ ( δ, δ) 3 ≡ ∈ − a (θ) 0 for θ π δ, π + δ 3 ≡ ∈ 3 − 3 0 a (θ) 3 for θ [0, π ] ≤ 3 ≤ ∈ 3
π 0 a′ (θ) 5 for θ [0, ]. ≥ 3 ≥ − ∈ 3 One easily checks that this satisfies conditions (3.26) and (3.27) for l = 3. Furthermore, if we set, for l> 0,

l al := a1, a2, 3 a3 then al satisfies the conditions of (3.26) and (3.27) for the
(l, 3) bundle, and furthermore satisfies

0 a (θ) l, 0 a′ (θ) 5l. ≤ 3 ≤ ≥ 3 ≥− In ([SS92] Proposition 2.7) the Yang-Mills energy of these connections is computed, and takes the form π 2 2 3 2 2 2 F (a) 2 = π (a1′ ) G1 + (a1 + a2a3) /G1 + (a2′ ) G2 + (a2 + a1a3)/G2 (3.28) ∇ L Z0  2 2 +(a3′ ) G3 + (a3 + a1a2) /G3 d θ, where  G = f2f3 , G = f3f1 , G = f1f2 1 f1 2 f2 3 f3 f (θ)= 2sin π + θ , f (θ) = 2sin π θ , f (θ)= 2sin(θ) . 1 3 2 3 − 3 Note that some terms in the energy
formula involve factors
of the Gi which can blowup at one endpoint or the other, but the boundary conditions for a ensure that these are finite integrals. In particular, for our initial choice of a = a3, we obtain some value for the Yang-Mills energy, call it C. We furthermore observe that every term in (3.28) is at worst quadratic in a3 and a3′ , which both grow linearly with l, and hence it follows that there is a different constant C such that

2 2 F (a ) 2 Cl . ∇ l L ≤

As the Sadun–Segert connection is constructed by energy minimization within this symmetry class ([SS92] Proposition 3.4, Theorem 3.10), its energy must lie below that of this test connection, finishing the proof of (3.25).  20 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

4. The Index of a positive Einstein Metric Let X4 be a smooth, closed, four-dimensional manifold. Furthermore suppose g is a critical point for the normalized total scalar curvature functional given in (1.1):

1/2 S [g] = Vol(g)− Rg dVg, 4 ZX where Rg is the scalar curvature of g. It follows that g is an Einstein metric, whose Ricci tensor is given by 1 Ric(g)= 4 Rg (see [Bes87], Chapter 4C). To study the second variation of at g, one uses the splitting of the space of sections of the bundle of symmetric two-tensors (seeS [Sch06] for details). The stability operator, corresponding to transverse-traceless variations of g, is given by

( (h))ij =∆hij + 2Rikjℓhkℓ (4.1) L =∆h + 2W h 1 Rh . ij ikjℓ kℓ − 6 ij This defines an index form I(h, h)= h, (h) dV h L i (4.2) ZX = h 2 + 2W (h, h) 1 R h 2 dV, − |∇ | − 6 | | ZX where  

W (h, h)= Wikjℓhkℓhij. The index ı(g) of an Einstein metric is the number of positive eigenvalues of (equivalently, the number of negative eigenvalues of ). The nullity ν(g) of an Einstein metricL is the dimension of the kernel of , i.e., the dimension−L of the space of infinitesimal Einstein deformations (see Chapter 12 of [Bes87])L . With this background we can give the proof of Theorem 1.3. 2 4 2 4 2 4 Proof of Theorem 1.3. Note that : S0 (T ∗X ) S0 (T ∗X ), where S0 (T ∗X ) is the bundle of trace-free symmetric two-tensors.L It follows from→ ([Hui85], Lemma 3.4), that1 W (h, h) 2 W h 2. − ≥− √3 | || | Therefore,

h, (h) dV h 2 4 W h 2 + 1 R h 2 dV h −L i ≥ |∇ | − √3 | || | 6 | | ZX ZX   = h, ∆+ 1 R V h dV, h − 6 − i ZX where
V = 4 W . √3 | | Since dim(S2(T X4)) = 9, applying Theorem 2.1 to the operator = ∆+ 1 R V gives 0 ∗ N − 6 − W 2 dV 2 X | | (4.3) ı(g)+ ν(g) 1728e 4 2 . ≤ RY(X , [g])

1Note that in [Hui85], the norm of Weyl is the one induced by the metric on covariant 4-tensors, while we are using the norm of Weyl viewed as a section of End(Λ2). INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 21

Since g is Einstein, (4.4) Y(X4, [g]) = S [g]. Also, by the Chern–Gauss–Bonnet formula,

8π2χ(X4)= W 2 + 1 R2 dV | | 24 ZX
= W 2 dV+ 1 S [g]2. | | 24 ZX Substituting this into (4.3), using (4.4), and rearranging the inequality gives

χ(X4) S [g] 24π , ≤ s3+ δ [ı(g)+ ν(g)] 2 1 where δ = (24e )− , as required.  5. The proof of Theorem 1.4 Proof of Theorem 1.4. Let (X4, g) be an oriented four-manifold with positive scalar curvature. To obtain the estimate for the first Betti number we only need to make minor changes to the index estimate for Yang-Mills connections, since the Jacobi operator in the case of the trivial bundle is the Hodge Laplacian acting on Λ1. The only difference is the choice of conformal representative: in the trivial case, we use a Yamabe metric in the conformal class of g instead of the metric specified in Proposition 3.6. 1 1 Let 1 : Λ Λ denote the Hodge Laplacian. Then by the Hodge-de Rham theorem, 1 4H → 4 1 4 H (X , R) = ker 1, and dim ker 1 = b1(X ). Let ω H (X , R) be a harmonic one-form; by the classical BochnerH formula, H ∈

2 ω,ω 2 = ω + Ric(ω,ω) dV h−H1 iL |∇ | ZX
= ω 2 + 1 R ω 2 + Z(ω,ω) dV |∇ | 4 | | ZX
ω 2 + 1 R ω 2 + Z(ω,ω) dV . ≥ |∇ | 6 | | ZX Since Z is trace-free,
Z(ω,ω) √3 ω 2. ≥− 2 | | Therefore,

2 1 2 √3 2 1ω,ω L2 ω + 12 R ω 2 Z ω dV h−H i ≥ X |∇ | | | − | || | Z   = ∆+ 1 R V ω,ω , − 6 − L2 where
V = √3 Z . 2 | | Applying Theorem 2.1 to the operator ∆+ 1 R V with = Λ , we get − 6 − E 1 108e2 (5.1) b (X4) Z 2 dV . 1 ≤ Y(X4, [g])2 | | ZX Recall 4 σ (A ) dV 1 Z 2 + 1 R2 dV ρ (X4, [g]) = 2 g = X − 2 | | 24 . 1 Y(X4, [g])2 Y(X4, [g])2 R R
22 MATTHEWJ.GURSKY,CASEYLYNNKELLEHER,ANDJEFFREYSTREETS

Since g is a Yamabe metric,

R2 dV = Y(X4, [g])2. ZX Consequently,

Z 2 dV = 2ρ (X4, [g]) Y(X4, [g])2 + 1 R2 dV | | − 1 12 ZX ZX = 1 1 24ρ (X4, [g]) Y(X4, [g])2. 12 − 1 Substituting this into (5.1) gives (1.3).
+ 4 2 4 2 4 + 4 To estimate b (X ), let 2 : H (X ) H (X ) denote the Hodge Laplacian. Then b (X )= + + H → 2 dim ker 2 , where 2 is the restriction of 2 to Λ+, the bundle of self-dual two-forms. The space of self-dualH harmonicH two-forms is conformallyH invariant since the Hodge ⋆ operator is. Therefore, in estimating b+(X4) we are free to choose a conformal metric. If we take the bundle E to be the trivial bundle in Proposition 3.6, then there is a conformal metricg ˆ [g] and a t (0, 1] such that ∈ 0 ∈ (5.2) R = 2√6t W + . gˆ 0| gˆ | From now on we assume g =g ˆ. The operator + satisfies the Weitzenbock formula H2 + =∆+2W + 1 R, H2 − 3 + 2 2 2 where ∆ is the rough Laplacian. Since W : Λ+ Λ+ is trace-free and dim Λ+ = 3, we have the sharp inequality → W +(ω,ω) 2 W + ω 2. | |≤ √6 | || | Therefore,

+ 2 + 1 2 ω,ω 2 = ω 2W (ω,ω)+ R ω dV h−H2 iL |∇ | − 3 | | ZX
ω 2 4 W + ω 2 + 1 R ω 2 dV √6 3 ≥ X |∇ | − | || | | | Z   = ω 2 + 1 R ω 2 + 1 R 4 W + ω 2 dV . 6 6 √6 X |∇ | | | − | | | | Z     Using (5.2),

+ 2 1 2 √6 + 2 2 ω,ω L2 ω + 6 R ω 3 (2 t0) W ω dV h−H i ≥ X |∇ | | | − − | || | Z   ∆+ 1 R V ω,ω , ≥ − 6 − L2 where
V = √6 (2 t ) W + . 3 − 0 | | Applying Theorem 2.1 to the operator ∆+ 1 R V with = Λ+, we get − 6 − E 2 2 4 72e 2 + 2 b + (X ) (2 t0) W 2 (5.3) ≤ Y(X4, [g])2 − k kL = 3e2(2 t )2ρ (X4, [g]), − 0 + where ρ+ is given by (1.2). By (3.12) of Proposition 3.6, 1/2 ρ− t 1, + ≤ 0 ≤ INDEX-ENERGY ESTIMATES FOR YANG–MILLS CONNECTIONS AND EINSTEIN METRICS 23 hence 2 1/2 2 1/2 2 (2 t ) ρ (2 ρ− ) ρ (2ρ 1) . − 0 + ≤ − + + ≤ + − Substituting this into (5.3) gives (1.4). 

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Department of Mathematics, 255 Hurley Bldg, University of Notre Dame, Notre Dame, IN 46556 E-mail address: [email protected]

Department of Mathematics Princeton University, Princeton, New Jersey, 08540 E-mail address: [email protected]

Department of Mathematics Rowland Hall, University of California, Irvine, Irvine, 92697 E-mail address: [email protected]