aa r X i v : . [ m a t h . DG ] M a r AREA, SCALAR CURVATURE, AND HYPERBOLIC3-MANIFOLDS
BEN LOWE
Abstract.
Let M be a closed hyperbolic 3-manifold that admitsno infinitesimal conformally-flat deformations. Examples of suchmanifolds were constructed by Kapovich. Then if g is a Riemann-ian metric on M with scalar curvature greater than or equal to −
6, we find lower bounds for the areas of stable immersed minimalsurfaces Σ in M . Our bounds improve the closer Σ is to beinghomotopic to a totally geodesic surface in the hyperbolic metric.We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfacesin ( M, g ). We show this functional to be uniquely maximized, overall metrics of scalar curvature greater than or equal to −
6, by thehyperbolic metric. Our proofs use the Ricci flow with surgery. Introduction − − πχ (Σ) /
3. Nunes proved that equality impliesthat the metric splits as a Riemannian product of a constant negativecurvature metric on a surface and an interval [Nun13]. In this paperwe improve these results for surfaces in closed hyperbolic 3-manifoldsthat admit no infinitesimal conformally-flat deformations.1.2. Let M be a closed hyperbolic 3-manifold and let Σ be a closedimmersed surface in M . Let F Σ denote the set of all immersed surfacesΣ ′ in M homotopic to the inclusion of Σ in M — i.e., for which there xists a map F : Σ × [0 , → M such that F | Σ ×{ } ) is the inclusion ofΣ in M and F (Σ × { } ) = Σ ′ . For a Riemannian metric g on M , define A Σ ( M, g ) = inf { area(Σ , g ) : Σ ∈ F Σ } . It follows from [KM12b], [KM12a] and [Sep16] that every closed hy-perbolic 3-manifold contains lots of closed immersed π -injective min-imal surfaces whose principal curvatures are pointwise as small as de-sired. Fix such a surface Σ, and set ǫ Σ = 1Area g hyp (Σ) Z Σ | A | , where A is the second fundamental form of Σ. The previous versionof this paper contained an incorrect proof of the following statement,which we now state as a conjecture. Conjecture 1.1.
Suppose that the scalar curvature of g is greater thanor equal to − . Then for every δ > there exists ǫ (depending on g )such that if ǫ Σ < ǫ , then (1.2) A Σ ( M, g ) ≥ (1 − δ ) A Σ ( M, g hyp ) . We say that (
M, g hyp ) admits no infinitesimal conformal deforma-tions if the cohomology group(1.3) H ( π ( M ) , Ad)vanishes, where Ad is the Adjoint representation of π ( M ) ⊂ SO (3 , so (4 ,
1) via the inclusion so (3 , ֒ → so (4 , g hyp through conformally flatmetrics. Kapovich gave infinitely many examples of closed hyperbolic3-manifolds for which (1.3) vanishes [Kap94]. These are obtained byDehn surgeries on hyperbolic 2-bridge knots. (A knot is 2-bridge if itcan be realized by an embedding in R with only two local maxima.The figure-eight knot is an example) Theorem 1.4.
Conjecture 1.1 is true if ( M, g hyp ) admits no infinites-imal conformal deformations. M based on an asymptotic count of minimalsurfaces homotopic to immersed almost totally geodesic surfaces in thehyperbolic metric, and showed that the hyperbolic metric uniquelyminimizes this functional over all metrics with sectional curvature atmost − o state their result, we need to introduce some terminology. Let S ( M ) denote the set of subgroups of π ( M ) isomorphic to the fun-damental group of a closed surface, or surface subgroups , up to theequivalence relation of conjugacy in π ( M ). The limit set of a surfacesubgroup π (Σ) of π ( M ) is the set of accumulation points in ∂ ∞ H ofany orbit of the action of π (Σ) on H .A homeomorphism f : S → S is K-quasiconformal if for any ball B ( x, r ) ⊂ S there exists r ′ > B ( f ( x ) , r ′ ) ⊂ f ( B ( x, r )) ⊂ B ( f ( x ) , Kr ′ ). Here B ( x, s ) denotes the ball centered at x of radius s in the round metric on S . A K-quasicircle is a Jordan curve that isthe image of a round circle under a K-quasiconformal map S → S .Kahn-Markovic showed the existence of surface subgroups of π ( M )whose limit sets in ∂ ∞ H ∼ = S are K -quasicircles for K ց
1. It will beimportant for us that a surface subgroup of π ( M ) whose limit set isa (1 + ǫ )-quasicircle is realized by the injective inclusion of a minimalsurface with principal curvatures bounded above by some universalconstant times log(1 + ǫ ) [Sep16].Denote by S ǫ ( M ) the elements of S ( M ) whose limit sets are 1 + ǫ -quasicircles. Elements of S ( M ) are in one-to-one correspondence withhomotopy classes of π -injective immersed surfaces in M . For a metric g on M and for S ∈ S ( M ), we denote by Area g ( S ) the infimal areawith respect to g of a surface in the corresponding homotopy class.Calegari-Marques-Neves defined the following “entropy” functional onmetrics g : E ( g ) := lim ǫ → lim L →∞ inf log( g ( S ) ≤ π ( L −
1) : S ∈ S ǫ ( M )) L log L .
Their main result is the following:
Theorem 1.5 (Calegari-Marques-Neves) . If the sectional curvature of g is less than or equal to − , then E ( g ) ≥ E ( g hyp ) = 2 with equality if and only if g is isometric to the hyperbolic metric g hyp . Using Theorem 1.4, we can prove an analogue of their theorem for scalarcurvature, but with a curvature bound in the opposite direction. Thefirst version of this paper contained an incorrect proof of the followingstatement, which we now state as a conjecture.
Conjecture 1.6.
If the scalar curvature of g is greater than or equalto − , then E ( g ) ≤ E ( g hyp ) . with equality if and only if g is isometric to g hyp . s above we are are able to prove it in a special case. Theorem 1.7.
Conjecture 1.6 is true if ( M, g hyp ) has no infinitesimalconformally flat deformations. M backto the hyperbolic metric, up to rescaling, after finitely many surgeries.An important point to check is that the minimal surfaces we consideravoid the surgery regions. The regions that are removed in surgery re-semble long thin tusks capped at the end by a (potentially much larger)region homeomorphic to a ball. A simple argument using the mono-tonicity formula guarantees that the minimal surfaces we consider donot enter these regions. To finish the proof we use the fact that nor-malized Ricci flow takes small perturbations of the constant curvaturemetric back to the constant curvature metric at an exponential rateof convergence. The exponent for the convergence is determined bythe linearization of the RHS of the normalized Ricci flow we consider.We use the work of Knopf-Young [KY09] and the assumption that M admits no infinitesimal conformally flat deformations to bound thespectrum of the linearization and control the rate of convergence to thehyperbolic metric.1.5. The strategy of obtaining bounds on areas of surfaces via Ricciflow is by no means original to this paper ( [BBEN10], [CM05], [MN12].)Conceivably other evolution equations for metrics could be used to simi-lar effect, although we only know of the paper by Ambrozio-Montezuma[AM18], which studied the Simon-Smith widths of conformal classes ofmetrics on S via Yamabe flow.The work of Bray-Brendle-Eichmair-Neves [BBEN10] is the closestthing to a positive curvature analogue of the results here. For closed3-manifolds with a positive lower scalar curvature bound, they provesharp upper bounds for the area of an embedded projective plane.They also show that the case of equality implies that the 3-manifold inquestion had to be isometric to the standard RP . Unlike this paper,however, they only use Ricci flow in characterizing the case of equality,where moreover they only use the short-time existence.The paper by Marques-Neves [MN12] on the other hand, which ob-tained sharp area bounds for min-max minimal surfaces in 3-manifoldswith non-negative Ricci curvature under positive scalar curvature lowerbounds, did use the long-time behavior of Ricci flow. (But see [Son18],which removes the non-negative Ricci curvature assumption and uses nly the short-time existence.) Both [BBEN10] and [MN12] were im-portant sources of inspiration for this paper.1.6. We now describe some other results related to this paper. In[And06], Anderson gives an argument to show how Perelman’s workallows one to compute the Yamabe invariant of a hyperbolic 3-manifold M . As in this paper, he starts with a general metric and evolves it un-der Ricci flow with surgery while keeping track of how scalar curvatureand the relevant geometric quantities are changing. A simple conse-quence of the computation of the Yamabe invariant is that any metricon M that is not hyperbolic and that has scalar curvature greater thanor equal to that of the hyperbolic metric must have volume greater thanthat of the hyperbolic metric. (This is actually conjectured to hold inall dimensions; see [Gut11] for further discussion and some results inthat direction.)A landmark comparison theorem for negative scalar curvature lowerbounds is the hyperbolic positive mass theorem, proved by Min-Oo inthe spin case and Andersson-Cai-Galloway in dimension less than 8.It implies as a special case that a metric on H n with scalar curvaturegreater than or equal to − n ( n −
1) that is equal to the standard hyper-bolic metric outside of a compact set must actually be isometric to thestandard H n . This special case also follows from recent work of Li [Li],who proved a comparison theorem for polyhedra in manifolds with anegative lower scalar curvature bound.Finally we mention the paper by Ache-Viaclovsky [AV15], whichalso considers hyperbolic 3-manifolds that admit no infinitesimal con-formally flat deformations in a geometric analysis context.1.7. Acknowledgments.
I thank Chao Li for a helpful conversationand for telling me about the Bamler paper [Bam17]. I thank AntoineSong for a helpful conversation. I also thank my advisor FernandoCoda Marques for a helpful conversation related to this paper and forhis support. I am indebted to Andre Neves for bringing to my attentionan error in the first version of this paper and for his efforts to help mefix it. 2.
Proof of Theorem 1.4
Let M be a closed hyperbolic 3-manifold and let Σ be a closed im-mersed π -injective surface in M . Let F Σ denote the set of all immersedsurfaces in M homotopic to the inclusion of Σ in M . For a Riemannianmetric g on M , define A Σ ( M, g ) = inf { Area g (Σ) : Σ ∈ F Σ } . n this section we will first prove Theorem 1.4 in the special casethat Σ is totally geodesic in ( M, g hyp ). For (
M, g hyp ) to satisfy thehypothesis of infinitesimal conformal rigidity, Σ must necessarily failto be embedded [JM87]. The author does not know whether thereare infinitesimally conformally rigid closed hyperbolic 3-manifolds thatcontain immersed totally geodesic surfaces, so this case might well bevacuous. We give the proof of this case, however, since it is illustrativeand contains all of the ideas for the proof of the general case of Σalmost totally geodesic. In this special case we are also able to provethat g is isometric to g hyp in the event that equality in the inequalityis realized and A Σ ( M, g ) = − πχ (Σ) = A Σ ( M, g hyp ) , which anticipates the rigidity in the case of equality in Theorem 1.7.First, we claim that we can assume that Σ is two-sided (has trivialnormal bundle in M .) If Σ is not two-sided, then we can find animmersion of a double cover Σ ′ of Σ that is. The claim then followsfrom the fact that homotopies of Σ lift to homotopies of Σ ′ and thearea of an immersion of Σ is half that of the immersion of Σ ′ that liftsit. From now on we assume that Σ is two-sided.The proof will be based on the Ricci flow with surgery. By work ofHamilton-Perelman, we know that Ricci flow with surgery starting at( M, g ) converges to the hyperbolic metric after rescaling the metricsto have volume that of the hyperbolic metric. To prove the inequal-ity, we study how A Σ ( M, g ) evolves under Ricci flow in comparison to A Σ ( M, g hyp ). Proof.
There exists a real number T and a family of metrics g ( t ) suchthat g ( t ) satisfies the Ricci flow equation(2.1) ∂∂t g ( t ) = − g ( t ) for t ∈ [0 , T ) ( [DeT83], [Ham82].) It will be important for us that thescalar curvature satisfies the evolution equation(2.2) ∂∂t R g ( t ) = ∆ R g ( t ) + 2 | Ric g ( t ) | . If we set g = g hyp , then g hyp ( t ) = (1 + 4 t ) g hyp solves (2.1) with T = ∞ . If T < ∞ , then high curvature regions develop as t ր T . Wecan perform surgery to cut out the high curvature regions in a standardand controlled way, so that bounds on all relevant geometric quantitiescontinue to hold after the surgery, and we can then restart Ricci flowand repeat. The work of Perelman implies that this can be done in such way that only finitely many surgeries occur total, which we explain inmore detail below. In the case of our ( M, g ), which cannot be writtenas a non-trivial connect sum, every surgery removes an inessential neckbounding a 3-ball.2.1. Σ
Totally Geodesic Case 1: No Surgeries.
To start out, we’regoing to give the proof for Riemannian metrics g such that there exist { g ( t ) : t ∈ [0 , ∞ ) } that satisfy (2.1) for all time and converge to thehyperbolic metric on M after rescaling the metrics to have volume thatof the hyperbolic metric.Let Σ be a closed immersed totally geodesic surface in ( M, g hyp ).We take Σ t to be a closed immersed minimal surface in ( M, g ( t )) thatminimizes g ( t )-area over all surfaces homotopic to the inclusion of Σin M . There exists at least one such surface by [SU82] or [SY79]. Forfixed t and Area Σ t ( s ) the area of Σ t in the metric g ( s ), we have that(2.3) dds (Area Σ t ( s )) | s = t = − Z Σ t Ric g ( t ) ( E , E ) + Ric g ( t ) ( E , E ) dµ g ( t ) , where ( E , E ) is a local orthonormal frame on Σ t for g ( t ) and dµ g ( t ) is the area form for the metric on Σ t induced by g ( t ). Recall thatwe assume by the comment following Theorem 1.4 that Σ is two-sided.If ν is the unit normal vector to Σ we can write the integrand of theabove equation as R ( E , ν, E , ν ) + R ( E , ν, E , ν ) + 2 R ( E , E , E , E )= 12 R + R ( E , E , E , E )= 12 R + K + 12 | A | , where R ( · , · , · , · ) and R are respectively the curvature tensor and thescalar curvature of ( M, g ( t )), K is the Gauss curvature of Σ t , and A is the second fundamental form of Σ. To get the last equality, we’reusing the Gauss equation and the fact that Σ t is minimal. UsingGauss-Bonnet and the fact that | A | ≥
0, we then have that(2.4) dds (Area Σ t ( s )) | s = t ≤ Z Σ t − Rdµ g ( t ) − πχ (Σ) . Let R min ( t ) be the minimum of the scalar curvature of g ( t ). Thensince R min (0) = −
6, we have by comparing with Ricci flow starting at g hyp and the parabolic maximum principle that(2.5) R min ( t ) ≥ − t + 1) . he inequalities (2.4) and (2.5) imply that(2.6) dds (Area Σ t ( s )) | s = t ≤ Σ t ( t )4 t + 1 − πχ (Σ) . Set A ( t ) = A Σ ( M, g ( t )). If g (0) = g hyp , we set A hyp ( t ) = A Σ ( M, g ( t )).By the same argument as the proof of Lemma 9 of [BBEN10], the func-tion A ( t ) is Lipschitz and thus differentiable almost everywhere. Forany fixed t , A ( t ) ≤ Area Σ t ( t ) . We also have that A ( t ) = Area Σ t ( t ) . If A ( t ) is differentiable at t , we therefore have A ′ ( t ) = Area ′ Σ t ( t ) . For almost every t , A ( t ) thus by (2.6) satisfies the inequality(2.7) A ′ ( t ) ≤ A ( t )4 t + 1 − πχ (Σ) . If the metric we started with was hyperbolic, then we would have that(2.8) A ′ hyp ( t ) = 3 A hyp ( t )4 t + 1 − πχ (Σ) . Therefore, since A (0) is less than or equal to A hyp (0), it follows bytaking the difference of (2.8) and (2.7) and solving explicitly the ODE y ′ ( t ) = 3 y ( t )4 t + 1obtained by replacing the inequality sign with an equality sign that(2.9) A hyp ( t ) − A ( t ) ≥ (4 t + 1) ( A hyp (0) − A (0)) . To get (2.9), we are using the fact that A is, as a Lipschitz function,absolutely continuous, and so the fundamental theorem of calculus canbe applied.As t tends to infinity, the metrics g ( t ) converge to the hyperbolicmetric when rescaled to have volume that of the hyperbolic metric.Denote these rescaled metrics by ˆ g ( t ). Then ˆ g ( t ) − g hyp can be madepointwise arbitrarily small in any C k norm by taking t large enough.Knopf-Young [KY09] consider a normalized Ricci flow, which theycall KNRF, defined by the following equation(2.10) ∂∂t g ( t ) = − g ( t ) − g ( t ) . Note that the hyperbolic metric is fixed by KNRF. Solutions toKNRF differ from solutions to usual Ricci flow by rescalings in space nd time. If ˜ g (˜ t ) is a solution to (2.10), then the time variable t for thecorresponding Ricci flow g ( t ) is equal to log(1 + 4˜ t ). For metrics g in a sufficiently small neighborhood U of the hyperbolic metric in the C ,α -norm, Knopf-Young show that the solution to (2.10) starting at g converges back to the hyperbolic metric exponential fast. They do soby bounding the spectrum of the linearization A g hyp at the hyperbolicmetric of the RHS of (2.10), which they compute to be(2.11) A g hyp ( h ) = ∆ h − Hg hyp + 2 h, where h is a symmetric (2 ,
0) tensor and H = g ij h ij is the trace of h . The linearized operator A g hyp is self-adjoint and elliptic with dis-crete spectrum contained in an interval ( −∞ , ω ]. If ω < g ( t ) isthe KNRF starting at g contained in the C ,α neighborhood describedabove, then the C norm g ( t ) − g hyp is bounded above by a constanttimes e ω t for any ω > ω . Using a Bochner formula for symmetric(2 ,
0) tensors due to Koiso, Knopf-Young show that ω can be taken tobe equal to − h be an eigentensor with eigenvalue −
1. Then it follows from[KY09][Section 5] that h must be trace-free and Codazzi, where recallthat a symmetric (2 ,
0) tensor T is Codazzi if its covariant derivativetensor is also symmetric– i.e., ∇ X T ( Y, Z ) = ∇ Y T ( X, Z ) . Trace-free Codazzi tensors over a hyperbolic 3-manifold correspondto infinitesimal conformally flat deformations of the hyperbolic met-ric (see, e.g., the introduction of [Bei97].) The space of infinitesimalconformally flat deformations is given by the cohomology group(2.12) H ( π ( M ) , Ad) , where Ad is the Adjoint representation of π ( M ) ⊂ SO (3 ,
1) on so (4 , so (3 , ֒ → so (4 , M there can be no eigentensor h as above with eigenvalue −
1, and so ω can be taken to be less than −
1. On the other hand, we point outthat if (2.12) is nonzero, which happens for example whenever (
M, g hyp )contains an embedded totally geodesic surface [JM87], then there willbe eigentensors with eigenvalue −
1, and so the argument below wouldfail.Now assume that (
M, g hyp ) admits no infinitesimal conformal defor-mations. Let ˆ g ( t ) be the rescaling of our initial Ricci flow g ( t ) to havevolume equal to that of g hyp at all times. Let ˜ g (˜ t ) be the KNRF with g (0) = g (0). Since ˜ g (˜ t ) C -converges to g hyp at a rate of e ω ˜ t for some ω < −
1, we have that ˆ g ( t ) C -converges to the hyperbolic metric ata rate of t ω/ , since ˜ t = log(1 + 4 t ). This is because after some fi-nite time ˆ g ( t ) will be contained in the C ,α neighborhood U of g hyp described above.Dividing Equation (2.9) by ( V ol ( g ( t )) /V ol ( g hyp )) / , which is lessthan or equal to a constant times t , we have that for some positiveconstant C A Σ ( M, g hyp ) − A Σ ( M, ˆ g ( t )) ≥ C t / . To obtain this inequality we have also used the fact that
V ol ( g ( t )) ≥ V ol ( g hyp ( t )), which follows from the theorem in [And06] described insubsection 1.6 of the introduction, since the scalar curvature of g ( t ) isgreater than that of g hyp ( t ) at all times t .But taking t large and using ˆ g ( t )’s convergence to the hyperbolicmetric at a rate of t ω/ gives a contradiction, since this implies that forsome constant C (2.13) |A Σ ( M, g hyp ) − A Σ ( M, ˆ g ( t )) | ≤ C t ω/ , and ω < − Case of Equality.
Now suppose that for some metric g withscalar curvature greater than or equal to − A Σ ( M, g ) = A Σ ( M, g hyp ) . Then we will show, following the argument for the equality case of themain result of [BBEN10], that g is isometric to g hyp . First, we run theRicci flow for a small interval of time [0 , ǫ ). We write the evolutionequation for scalar curvature under Ricci flow as(2.14) ∂∂t R g ( t ) = ∆ R g ( t ) + 23 R g ( t ) + 2 | ˚ Ric g ( t ) | . By (2.9), A Σ ( M, g ( t )) ≤ A Σ ( M, g hyp ( t )) for t ∈ [0 , ǫ ), so by what wehave shown above we must have that the minimum of R g ( t ) is identicallyequal to − t on [0 , ǫ ). Consequently, by the strong maximum principle, R g ( t ) is identically equal to − t which implies that ˚ Ric g ( t ) is identicallyzero on [0 , ǫ ). The metric g must then have been Einstein and, sincewe are in three dimensions, have had constant sectional curvature − − M isunique up to isometry, this implies that g is isometric to g hyp . (cid:3) .2. Σ Totally Geodesic Case 2: Finitely Many Surgeries.
Wenow modify the proof of the previous case in the event that Ricci flow(
M, g ( t )) starting from the initial metric develops singularities. Firstwe describe the general strategy. At a singularity, long tusks are form-ing that up to rescaling are nearly isometric to ( S , g round ) × ( − ǫ , ǫ )along most of their length, for ǫ small. In performing surgery, we chopoff part of the tusk. In order to implement the approach of the previ-ous case, we need to rule out the possibility that an area-minimizingsurface in the homotopy class of Σ enters the part of the tusk we chopoff in surgery, which we accomplish by using the monotonicity formula.This proves that A Σ ( M, g ( t )) is Lipschitz continuous at the surgerytimes. Bounds on all relevant geometric quantities— most importantlyfor us the pointwise lower bound on the scalar curvature— continue tohold after surgery, so from here the argument can proceed as in thefirst case.We now explain Bamler’s set-up, which we copy from his paper[Bam17] nearly verbatim. We don’t actually use any of his resultsfrom that paper, only his definitions and formulations of Perelman’sresults, which are concise and convenient for our purposes. To start wegive the definition of Ricci flow with surgery that we will use. Definition 2.15 (Ricci flow with surgery) . Consider a time interval I ⊂ R . Let T < T < ... be times in the interior of I which form apossibly infinite but discrete subset of R and divide I into the intervals I = I ∩ ( −∞ , T ) , I = [ T , T ) , ... and I k +1 = I ∩ [ T k , ∞ ) if there are only finitely many T i ’s. Con-sider Ricci flows ( M × I , g t ), ( M × I , g t ),... on 3-manifolds M , M ,... Assume that the metrics g it converge smoothly as t ր T i to aRiemannian metric g T i on M , and let U i − ⊂ M i and U i + ⊂ M i +1 be open subsets such that there are isometriesΦ i : ( U i − , g iT i ) → ( U i + , g i +1 T i ) , (Φ i ) ∗ g i +1 T i | U i + = g iT i | U i − . We assume that we never have U i − = M i and U i + = M i +1 . We alsoassume that every component of M i +1 contains a point of U i + . We call M = (( T i ) i , ( M i × I i , g it ) i , ( U i ± ) i , (Φ i ) i ) a Ricci flow with surgery on thetime interval I , and we call T , T ,... surgery times.If t ∈ I i , then ( M ( t ) , g ( t )) = ( M i × { t } , g it ) is called the time- t slice of M . For t = T i , we define the (presurgery) time T i − -slice to be( M ( T i − ) , g ( T i − )) = ( M i ×{ T i } , g iT i ). The points M i ×{ T i }\ U i − ×{ T i } are called presurgery points and the points M i +1 × { T i } \ U i + × { T i } re called surgery points. We will call a point that is not a presurgerypoint a non-presurgery point. Remark . This definition of Ricci flow with surgery is a slight spe-cialization of the one from [Bam17], since the only surgeries that occurfor our M pinch off a topologically trivial capped neck. Definition 2.17 ( ǫ -neck) . Let ǫ >
0, and consider U ⊂ ( M, g ). Thenwe say U is an ǫ -neck if there is a diffeomorphism Φ : S × ( − ǫ , ǫ ) → U and a λ > | λ − Φ ∗ g − g S × ( − ǫ , ǫ ) | C ⌈ ǫ − ⌉ < ǫ , where g S × ( − ǫ , ǫ ) is the standard metric on S × ( − ǫ , ǫ ) with constant scalar curvature2. We say that x is a center of U if x ∈ Φ( S × g on M and any ǫ > g ( t ) starting at g and definedfor all time that satisfies the following. At every surgery time T i , thecomponents of M ( T i − ) \ U i − are homeomorphic to D , and the subset M ( T i ) \ U i + is a disjoint union D i ∪ .. ∪ D im i of homeomorphic copies of D . Moreover, for every D ij , we can take the points on the boundary of U i − in M ( T i − ) corresponding to ∂D ij to be centers of ǫ -necks for the ǫ chosen at the start. This follows from Proposition 3.4 of [Bam17], thedefinition of Ricci flow with surgery with δ ( t )-precise cutoff, and thefact that we can take δ ( t ) to be pointwise smaller than any positivenumber (in this case, ǫ .)Moreover, for our specific M only finitely many surgeries occur. Thisis because the g ( t ), normalized to have volume that of the hyperbolicmetric, smoothly converge to the hyperbolic metric as t tends to infin-ity. This is explained in [And06][pg. 132] and [Cal19][pg. 60]. Sinceonly finitely many surgeries occur on any finite time interval, and nosurgeries occur when the normalized metric is sufficiently close to thehyperbolic metric, only finitely many surgeries occur total.Let Σ T i − be an immersed minimal surface in M ( T i − ) = ( M, g T i − )that realizes the minimum g T i − -area over all immersed surfaces homo-topic to Σ. We claim that Σ T i − does not intersect M ( T i − ) \ U i − , whichis diffeomorphic to a union of balls, provided that the ǫ chosen at thestart was taken sufficiently small. Any point p on the boundary of thisregion is the center of an ǫ -neck N .Φ : S × ( − ǫ , ǫ ) → N such that p ∈ Φ( S × { } ) and for some λ > | λ − Φ ∗ g T i − − g S × ( − ǫ , ǫ ) | C ⌈ ǫ − ⌉ < ǫ. ow assume for contradiction that Σ T i − passes through p . It cannotbe the case that Σ T i − is contained in N ∪ M ( T i − ) \ U i − , so Σ T i − mustintersect Φ( S × { x } ), for x slightly greater than − ǫ . We can perturbΦ( S × { x } ) slightly to an embedded sphere S that intersects Σ T i − transversely in a union of circles. These circles necessarily bound disksin Σ T i − , so let D p be a disk or an annulus in Σ T i − that passes through p . On the one hand, ∂D p can be filled in by a region in S with area atmost roughly 4 πλ . On the other hand, D p must intersect every cross-section Φ( S × { y } ) for − ǫ < y <
0. By the monotonicity formula,there is a universal constant c such that for y ∈ ( − ǫ + 1 / , − / D p ∩ Φ( S × ( y − / , y + 1 / > cλ . It follows by choosing disjoint unit intervals in ( − ǫ + 1 / , − /
2) thatif we chose ǫ such that ǫ is greater than 5 π/c , then the area of D p willbe larger than (4 π + 1) λ . By cutting out D p and gluing in a region of S we could then produce a surface homotopic to Σ T i − but with smallerarea, which is a contradiction.Since U i − is unchanged by surgery and we have an isometry U i − → U i + ,the same argument shows that a Σ T i + that minimizes area over surfaceshomotopic to Σ in the surgered manifold M T i + is contained in U i + . Thisproves that the function A ( t ), defined as the minimum area of a surfacehomotopic to Σ in M ( t ), is well-defined. Since it is also Lipschitz, andsince the estimate (2.5) for the minimum of scalar curvature also holdsfor Ricci flow with surgery, the arguments of the previous section apply.This proves Theorem 1.4 in the case that Σ is totally geodesic in thehyperbolic metric, in the event that singularities occur in Ricci flowstarting at the initial metric.2.3. Proof for General Σ . Fix a metric g with scalar curvaturegreater than or equal to −
6, and let g ( t ) be a Ricci flow with surgerystarting from g . The inequality in Theorem 1.4 can be rearranged to(2.19) A Σ ( M, g hyp ) − A Σ ( M, g ) ≤ δ A Σ ( M, g hyp ) . Fixing a surface Σ as in the theorem and with notation as above, weassume that for some δ > A hyp (0) − A (0) > δ A hyp (0) . Since Σ is not totally geodesic, in place of (2.8) we have the equality(2.20) A ′ hyp ( t ) = (1 − ǫ Σ ) (cid:18) A hyp ( t )4 t + 1 (cid:19) − πχ (Σ) . here recall that ǫ Σ is the average of | A | over Σ. Taking the differencewith (2.7), we obtain(2.21) ( A hyp ( t ) − A ( t )) ′ ≥ (cid:18) A hyp ( t ) − A ( t ))4 t + 1 (cid:19) − ǫ Σ t + 1 ( A hyp ( t )) . Since A hyp ( t ) = (1 + 4 t ) A hyp (0), integrating gives A hyp ( t ) − A ( t ) ≥ (4 t + 1) ( A hyp (0) − A (0)) − ǫ Σ A hyp (0) t (2.22) > (4 t + 1) ( δ A hyp (0)) − ǫ Σ A hyp (0) t. (2.23)Dividing by ( V ol ( g ( t )) /V ol ( g hyp )) / as above, which grows at mostlinearly, and using the fact that ˆ g ( t ) converges to g hyp at a rate of atleast t ω for ω < − t tends to infinity, which implies the inequality(2.13), we obtain a contradiction if ǫ Σ is sufficiently small. This showsthat for every δ >
0, (2.19) holds for ǫ Σ sufficiently small.3. Counting with a Lower Scalar Curvature Bound
In this section, we prove Theorem 1.7. Assume that (
M, g ) is asin Theorem 1.7, with scalar curvature greater than or equal to − S ǫ ( M ) is realized by an immersed g hyp -minimal surface which has principal curvatures at most C log(1 + ǫ ), for some universal constant C [Sep16]. For every δ >
0, Theorem1.4 implies that for ǫ sufficiently small(3.1) { Area g ( S ) ≤ π ( L −
1) : S ∈ S ǫ ( M ) }≤ { Area g hyp ( S ) ≤ π ( L − / (1 − δ ) : S ∈ S ǫ ( M ) } . Dividing by L log L , sending L to infinity, and sending δ to 0 provesthat E ( g ) ≤ E ( g hyp ).Now assume E ( g ) = E ( g hyp ). We follow the same strategy as in2.1.1. Let g ( t ) be a Ricci flow starting at g for t contained in a smallinterval of time [0 , ǫ ′ ). We claim that E ( g ( t )) ≥ E ( g hyp ( t )), where g hyp ( t ) is the Ricci flow starting at g hyp . To see this, we claim that forevery δ > g ( t ) ( S ) ≤ (1 + 4 t ) δ Area g ( S ) , provided ǫ was taken sufficiently small and S ∈ S ǫ ( M ). We have thatfor any δ ′ >
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