ALMOST-SCHUR LEMMA CAMILLO DE LELLIS AND PETER M. TOPPING

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1 ALOST-SCHU LEA CAILLO DE LELLIS AND PETE. TOPPING. Itroductio Schur s lemma states that every Eistei maifold of dimesio 3 has costat scalar curvature. Here, g) is defied to be Eistei if its traceless icci tesor ic := ic g is idetically zero. I this short ote we ask to what extet the scalar curvature is costat if the traceless icci tesor is assumed to be small rather tha idetically zero. Theorem.1. For ay iteger 3, if, g) is a closed iemaia maifold of dimesio with weakly positive icci curvature, the ) 4 1) ic.1) ) where is the average value of over. Sice ic g we immediately get: = ic + 1 ),.) Corollary.. Uder the same coditios as i the theorem, ic g ic ) g.3) These estimates are sharp i the followig seses. First, the costats are the best possible because if we were to reduce either costat the iequalities would fail for certai small but high-frequecy deformatios of the roud sphere as we discuss i Sectio 1. Ideed, if g is the metric of the roud sphere the we ca take a coformal deformatio 1 + f)g where f is a eigefuctio of the Laplacia o the sphere correspodig to a suitably large eigevalue. Secod, the curvature coditio ic caot simply be dropped, as we discuss i Sectio : For 5, we show that ay such iequality the fails 1

2 CAILLO DE LELLIS AND PETE. TOPPING eve if we restrict to be the sphere S. For example, we ca fid metrics g o S which make the ratio of the left-had side of.3) to the right-had side of.3) arbitrarily large. If we are able to prescribe the topology of, the the same thig ca be egieered eve i dimesio = 3: we ca fid maifolds 3, g) which make the same ratio arbitrarily large. We leave ope the possibility that iequalities of this form may hold for = 3 ad = 4 with costats depedig o the topology of. Of all kow iequalities which geeralise a exact statemet such as Schur s lemma above, perhaps the closest oe to our result is the iequality of üller ad the first author [1] which geeralises the wellkow assertio that the oly totally umbilic closed hypersurfaces of Euclidea space are spheres. I fact, we observe that our method also gives that result with the sharp costat for covex hypersurfaces of ay dimesio, eve withi more geeral Eistei ambiet maifolds. Details, ad related results, will appear i []. Proof. ecall that the cotracted secod Biachi idetity tells us that δic + 1 d = where δic) j := i ij ) ad hece that ic = d..4) δ Let f : be the uique solutio to f = f =. We may the compute ) ) = f = = = δ ic, df = d, df ic, Hessf f g ic L Hessf f g L. ic, Hessf.5).6) Now by itegratio by parts i.e. the Bocher formula) we kow that Hessf = f) ic f, f).7)

3 ad therefore Hessf f g = ALOST-SCHU LEA 3 = 1 = 1 Hessf 1 f) f) ic f, f) ) ic f, f),.8) ad sice the icci curvature is weakly positive, we have ) 1 1 Hessf f g L ),.9) which ca be combied with.6) to give.1). emark.3. We ote that the icci term which we throw away i the proof does ot destroy optimality because that term is lower order i.e. oly ivolves first derivatives ad is thus isigificat for very high frequecy f. emark.4. We oly use the icci hypothesis i the proof i order to obtai the L estimate Hessf f)..1) oreover, a slight adaptatio of the proof would establish a L p versio of our results o ay maifold supportig a Caldero-Zygmud iequality Hessf p C f) p..11) 1. Secod variatio argumets We will show that the costats i.1) ad.3) are optimal. We do this by computig the secod variatio formula of each side of the iequalities based at the roud sphere of dimesio 3. If the costat i either iequality were reduced at all, the we could fid small, high-frequecy perturbatios of the roud sphere which violated both estimates. Optimality of.1) ad.3). First of all observe that, by.), the optimality of oe iequality implies the optimality of the other. We ext cosider the stadard sphere = S, σ) for which ic = 1)σ ad = 1), ad deform it through a oe-parameter family of

4 4 CAILLO DE LELLIS AND PETE. TOPPING iemaia maifolds t = S, g t ) where g t = 1 + tf)σ. We assume that f C ) ad f =. Set F t) := C ic g t = C 1) ic C t ic g + 1 V ) =: C 1)F 1 t) C F t) + 1 F 3t) 1.1) where V is the volume of t. We write dvol for the volume elemet. Straightforward calculatios see for istace Sectio.3.1 of [4]) give t dvol = f dvol 1.) d dt V = 1.3) t g ij = fσ ij 1.4) t ic ij = 1 fσ ij + )f ;ij ) 1.5) t = 1) f 1)f 1.6) d dt = 1.7) Therefore F ) =. We ext show that, for ay costat C < ), there is a choice of f such that F ) <. This will imply the optimality of.3) as desired. We start by remarkig that F ) = d ) t + t dvol dt = 1) t + t ) +4 1) t t dvol + 1) t dvol = 1) d + t ) 1) d V dt.1.8)

5 Similarly, F 3 ) = d dt ALOST-SCHU LEA 5 1 V dv dt = 1) d V + d dt V ) d dt } {{ } = by 1.7) = 1) d V Fially we compute F 1 ) = t ic + Note that ) + V dv d d dt 1 ) ) ) dt dt V }{{} = by 1.3) + 1) d d + 1) t ic t dvol + t ic = t ic ij ic kl g ik g jl + ic ij ic kl t g ik g jl. 1.9) ic t dvol. 1.1) = 1) t. 1.11) t ic = t [ t ic ij g jl ) iciα g αl] Therefore, we coclude F 1 ) = 1) = 1) t + [ t icij g jl) t iclα g αi)] = 1) t 4 1)f t ic ij σ ij + t ic + 1) f 1.1) t + + 1) f + 4 1) + 1) t dvol = 1) d + + 1) t ic 4 1) t t dvol t ic 4 1) f t ic ij σ ij f t ic ij σ ij f 1) d V 1.13)

6 6 CAILLO DE LELLIS AND PETE. TOPPING Puttig together 1.13), 1.8) ad 1.9) we get F ) = C t ) + C 1) t ic 4C 1) 1) f t ic ij σ ij + C 1) 1) f.1.14) Next, we have t ) = 1) f t ic ij σ ij = 1) t ic = 4 = 1) 4 f) df + f )1.15) df 1.16) f) + f) ) + D f 4 f) ) 1) df.1.17) 4 where i the last lie we used the Bocher formula.7)). Assume ow that C = ) ε for some positive ε. Isertig 1.15), 1.16) ad 1.17) ito 1.14), we coclude F ) a)ε f) + b, ε) df + c, ε) f, 1.18) where the costat a is strictly positive sice 3). By choosig f to be a eigefuctio of the Laplacia with sufficietly large eigevalue, we the have F ) < as desired.. Couterexamples without the hypothesis ic. Our results assume we are workig o a maifold of weakly positive icci curvature. We ow wish to ask whe we have a hope of provig a iequality of the form ) C o more geeral maifolds, g). ic.1) Propositio.1. For ay C < ad iteger 5, there exists a metric g o the sphere S such that.1) fails. For smaller, we kow couterexamples oly whe the topology of is allowed to deped o C: Propositio.. For ay C <, there exists a closed 3-maifold, g) such that.1) fails.

7 ALOST-SCHU LEA 7 Proof. Propositio.1.) All we have to do is to coect two roud spheres of radii 1 ad, say, by a small eck. O the two spherical parts, the traceless icci tesor ic is zero. Therefore for ay C) we ca make the right-had side of.1) as small as desired for 5, sice by scalig dow the size of the eck, the itegral of ic over the eck will also be scaled dow to as small a value as we wish. eawhile, the differet radii of the spherical parts esure that the scalar curvature is differet o each sphere, ad thus the left-had side of.1) caot be small. Proof. Propositio..) This costructio is loosely related to the oe above. The basic buildig block is ay hyperbolic costat sectioal curvature 1) 3-maifold N, h) which fibres over the circle. A result of Thursto implies that if S is a closed surface of geus at least, the the 3-maifold arisig by gluig the boudary compoets of [, 1] S usig a pseudo-aosov diffeomorphism of the fibre S must admit a hyperbolic metric [3]). Let us write N m for the m-fold coverig of N obtaied by takig covers of the base circle, ad lift the metric h to a metric h o N m. We also pick a poit p i N ad ay oe poit p i each N m which projects to p uder the coverig. The idea the, for each m N, is to attach oe N m, h) to aother scaled copy N m, h) via a m-idepedet eck attached to small eighbourhoods of p i each N m, to give a ew maifold, g). With this costructio, the right-had side of.1) is idepedet of m, but the left-had side will icrease without boud as m at a asymptotically liear rate. Ackowledgemets. We thak Vlad arkovic for a useful coverstatio. CDL was supported by a grat of the Swiss Natioal Foudatio. PT was supported by The Leverhulme Trust. efereces [1] C. De Lellis ad S. üller, Optimal rigidity estimates for early umbilical surfaces. J. Differetial Geom. 69 5) [] D. Perez, PhD thesis, Uiversität Zürich. I preparatio. [3] W.P. Thursto, Hyperbolic structures o 3-maifolds II: Surface groups ad 3-maifolds which fibre over the circle. arxiv:math.gt/98145 [4] P.. Toppig, Lectures o the icci flow. L..S. Lecture ote series 35 C.U.P. 6) istitut für mathematik, uiversität zürich, witerthurerstrasse 19, CH-857 zürich, switzerlad.

8 8 CAILLO DE LELLIS AND PETE. TOPPING mathematics istitute, uiversity of warwick, covetry, CV4 7AL, uk.