Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds
aa r X i v : . [ m a t h . DG ] A p r RIGIDITY OF VOLUME-MINIMIZING HYPERSURFACES INRIEMANNIAN 5-MANIFOLDS
ABRA ˜AO MENDES
Abstract.
In this paper we generalize the main result of [4] for manifoldsthat are not necessarily Einstein. In fact, we obtain an upper bound for thevolume of a locally volume-minimizing closed hypersurface Σ of a Riemannian5-manifold M with scalar curvature bounded from below by a positive constantin terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturatesthe respective upper bound and M has nonnegative Ricci curvature, then Σ isisometric to S up to scaling and M splits in a neighborhood of Σ. Also, weobtain a rigidity result for the Riemannian cover of M when Σ minimizes thevolume in its homotopy class and saturates the upper bound. Introduction
A classical result due to Toponogov [15] says that the length of any closed simplegeodesic γ on a closed Riemannian surface M satisfies L ( γ ) inf M K ≤ π , where K is the Gaussian curvature of M . Furthermore, if equality holds, then M is isometric to the standard unit 2-sphere S ⊂ R up to scaling (see [9] for adifferent proof).A similar result could be imagined for minimal 2-spheres, instead of closed simplegeodesics, in dimension 3. But, it turns out that there is no area bound for minimal2-spheres in Riemannian 3-manifolds, as pointed out by Marques and Neves [10].Therefore, an extra hypothesis is needed.It is well known that if Σ is a stable minimal 2-sphere in a Riemannian 3-manifold M , then the area of Σ satisfies A (Σ) (cid:18) inf M R (cid:19) ≤ π, (1)where R is the scalar curvature of M . Moreover, if equality holds, then Σ is totallygeodesic and R is constant equal to inf M R on Σ. If we further assume that Σ islocally area-minimizing, then equality in (1) implies M to be isometric to ( − ε, ε ) × S up to scaling in a neighborhood of Σ, suposing that Σ is embedded in M . Thiscan be seen as a consequence of Bray, Brendle, and Neves’ work [5] (see [11] for analternative proof).In dimension n ≥
3, it is not difficult to construct manifolds Σ n with scalarcurvature R Σ ≥ α n >
0, for some constant α n depending only on n , and arbitrarily Date : April 27, 2018.The author is grateful to Fernando C. Marques, Marcos P. Cavalcante, Feliciano Vit´orio, andEzequiel Barbosa for their kind interest in this work. The author was financially supported byCAPES Foundation, Ministry of Education of Brazil. large volume. For example, consider Σ nr = S n − × S ( r ), where S n − ⊂ R n is thestandard unit ( n − S ( r ) ⊂ R is the circle of radius r >
0. Clearly, R Σ r = ( n − n −
2) and Vol(Σ r ) −→ ∞ as r −→ ∞ . However, these manifoldsare not diffeomorphic to S n .For the spherical case, Gromov and Lawson [7] developed a method which per-mits to construct metrics on Σ n = S n with scalar curvature R Σ ≥ n ( n −
1) andarbitrarily large volume if n ≥ n ≥ M n +1 = R × Σ n with Σ n as above.Bearing this in mind, Barros, Batista, Cruz, and Sousa [4] considered the caseof Einstein 4-manifolds embedded in Riemannian 5-manifolds which minimize thevolume in their homotopy classes. They proved: Theorem 1.1 (Barros-Batista-Cruz-Sousa) . Let M be a complete Riemannianmanifold with positive scalar curvature and nonnegative Ricci curvature. Supposethat Σ is a two-sided closed Einstein manifold embedded in M in such a way that Σ minimizes the volume in its homotopy class. Then, the volume of Σ satisfies Vol(Σ) / (cid:18) inf M R (cid:19) ≤ Vol( S ) / . (2) Moreover, if equality holds, then Σ is isometric to S , M is isometric to ( − ε, ε ) × S in a neighborhood of Σ , and the Riemannian cover of M is isometric to R × S , upto scaling. Our purpose in this work is to generalize Theorem 1.1 for manifolds that are notnecessarily Einstein. To do so, from the above comments, it is necessary an extraterm in (2). Our first result is the following:
Theorem 1.2 (Theorem 3.4) . Let M be a Riemannian manifold with scalar curva-ture R satisfying inf M R > and nonnegative Ricci curvature. If Σ is a two-sidedclosed hypersurface embedded in M which is locally volume-minimizing, then thevolume of Σ satisfies Vol(Σ) (cid:18) inf M R (cid:19) ≤ Vol( S ) + 112 Z Σ | ˚Ric Σ | dσ, (3) where ˚Ric Σ is the traceless Ricci tensor of Σ . Furthermore, if equality holds, then Σ is isometric to S and M is isometric to ( − ε, ε ) × S in a neighborhood of Σ , upto scaling. Our second result is the following:
Theorem 1.3 (Theorem 3.6) . Let M be a complete Riemannian manifold withscalar curvature R satisfying inf M R > and nonnegative Ricci curvature. Supposethat Σ is a two-sided closed manifold immersed in M in such a way that Σ minimizes the volume in its homotopy class. Then, the volume of Σ satisfies Vol(Σ) (cid:18) inf M R (cid:19) ≤ Vol( S ) + 112 Z Σ | ˚Ric Σ | dσ. (4) Moreover, if equality holds, then Σ is isometric to S and the Riemannian cover of M is isometric to R × S , up to scaling. IGIDITY OF VOLUME-MINIMIZING HYPERSURFACES IN 5-MANIFOLDS 3
Remark 1.4.
The covering map of Theorem 3.6 is explicit. In fact, it is given by G ( t, x ) = exp x ( tN ( x )), ( t, x ) ∈ R × Σ, where exp is the exponential map of M and N is a unit normal vector field defied on Σ.2. Preliminaries
In this section, we are going to present some terminologies and useful results.Let Σ n be a connected closed (compact without boundary) manifold of dimension n ≥
3. Denote by M (Σ) the set of all Riemannian metrics on Σ. The Einstein-Hilbert functional E : M (Σ) → R is defined by E ( g ) = R Σ R g dv g Vol(Σ n , g ) n − n , where R g is the scalar curvature of (Σ , g ). Denote by [ g ] = { e f g : f ∈ C ∞ (Σ) } the conformal class of g ∈ M (Σ). The Yamabe invariant of (Σ , [ g ]) is defined asthe following conformal invariant: Y (Σ , [ g ]) = inf ˜ g ∈ [ g ] E (˜ g ) . The classical solution of the Yamabe problem by Yamabe [17], Trudinger [16],Aubin [2] (se also [3]), and Schoen [14] says that every conformal class [ g ] containsmetrics ˆ g , called Yamabe metrics , which realize the minimum: E (ˆ g ) = Y (Σ , [ g ]) . Such metrics have constant scalar curvature given by R ˆ g = Y (Σ n , [ g ]) Vol(Σ n , ˆ g ) − n . Furthermore, Y (Σ n , [ g ]) ≤ Y ( S n , [ g can ])and equality holds if and only if (Σ n , g ) is conformally diffeomorphic to the standardunit n -sphere S n ⊂ R n +1 endued with the canonical metric g can . Therefore, as aconsequence of Obata’s theorem [13, Proposition 6.1], if Y (Σ n , [ g ]) = Y ( S n , [ g can ])and g has constant scalar curvature, then (Σ n , g ) is isometric to ( S n , g can ) up toscaling.When n = 4, a very useful tool is the Gauss-Bonnet-Chern formula for the Eulercharacteristic χ (Σ) of a closed Riemannian manifold (Σ , g ), which reads as follows:8 π χ (Σ) = Z Σ (cid:18) | W g | + 124 R g − | ˚Ric g | (cid:19) dσ, where W g and ˚Ric g = Ric g − ( R g /n ) g are the Weyl and the traceless Ricci tensorsof (Σ , g ), respectively.Before finishing this section, we are going to state two important inequalitiesproved by Gursky [8]. Theorem 2.1 (Gursky) . Let (Σ , g ) be a closed Riemannian manifold. If Σ hasnonnegative scalar curvature, then Z Σ | W g | dσ ≥ π ( χ (Σ) − ABRA˜AO MENDES and Y (Σ , [ g ]) ≥ (cid:18) π χ (Σ) − Z Σ | W g | dσ (cid:19) . (6) Remark 2.2.
Clearly, (5) and (6) are trivial if χ (Σ) ≤ χ (Σ) ≤
0, respectively.3.
The results
Let Σ be a closed hypersurface immersed in a Riemannian manifold M . Here,we suppose that Σ is two-sided , that is, there exists a unit normal vector field N defined on Σ. Proposition 3.1.
Let M be a Riemannian manifold with scalar curvature R sat-isfying inf M R > and Σ be a two-sided closed hypersurface immersed in M . If Σ is stable minimal in M , then the volume of Σ satisfies Vol(Σ) (cid:18) inf M R (cid:19) ≤ Vol( S ) + 112 Z Σ | ˚Ric Σ | dσ, (7) where ˚Ric Σ is the traceless Ricci tensor of Σ . Furthermore, if equality holds, then (i) Σ is isometric to S up to scaling, (ii) Σ is totally geodesic in M , (iii) Ric( N, N ) = 0 and R = inf M R on Σ ,where Ric is the Ricci tensor of M .Proof. Since the left hand side of (7) and R Σ | ˚Ric Σ | dσ are scaling invariant andinf M R >
0, without loss of generality, we may assume that inf M R = 12. Since Σis stable minimal, the stability inequality says that Z Σ ( |∇ f | − (Ric( N, N ) + | A | ) f ) dσ ≥ f ∈ C ∞ (Σ), where A is the second fundamental form of Σ in M . Taking f = 1 above and using the Gauss equation12 ( R − R Σ + | A | ) = Ric( N, N ) + | A | , (9)we have Z Σ ( R + | A | ) dσ ≤ Z Σ R Σ dσ, (10)where R Σ is the scalar curvature of Σ. Therefore, observing that R ≥
12 byhypothesis, it follows that12 Vol(Σ) ≤ Z Σ R Σ dσ ≤ Vol(Σ) / (cid:18)Z Σ R dσ (cid:19) / , (11)i.e., 12 Vol(Σ) ≤ Z Σ R dσ, (12)where above we have used the H¨older inequality.Now, let φ ∈ C ∞ (Σ), φ >
0, be the first eigenfunction of the stability operator L of Σ, L = ∆ + (Ric( N, N ) + | A | ) , (13) IGIDITY OF VOLUME-MINIMIZING HYPERSURFACES IN 5-MANIFOLDS 5 associated to the first eigenvalue λ = λ , that is, Lφ + λφ = 0 . (14)Because Σ is stable, λ ≥
0. Denote by g Σ the Riemannian metric on Σ inducedfrom M and define a new metric g = φ / g Σ . It is well known that the scalarcurvatures of g and g Σ are related according to the equations R g φ = − φ / ) + R Σ φ / = − φ − / ∆ φ + 43 φ − / |∇ φ | + R Σ φ / , which imply R g φ ≥ − φ − / ∆ φ + R Σ φ / . (15)Then, using (9), (13) and (14) into (15), we obtain R g φ ≥ φ / ( R + | A | + 2 λ ) ≥ φ / > , thus R g >
0. In particular, Y (Σ , [ g Σ ]) >
0. Denoting by W Σ the Weyl tensor of(Σ , g Σ ) and remembering that R Σ | W Σ | dσ is a conformal invariant of (Σ , g Σ ) indimension 4, it follows from Gursky’s theorem that Z Σ | W Σ | dσ ≥ π ( χ (Σ) − . (16)Then, using (12), (16) and the Gauss-Bonnet-Chern formula, we haveVol(Σ) ≤ (cid:18) Z Σ R dσ (cid:19) = 16 (cid:18) π χ (Σ) − Z Σ | W Σ | dσ + 12 Z Σ | ˚Ric Σ | dσ (cid:19) ≤ π + 112 Z Σ | ˚Ric Σ | dσ, which imply (7) because Vol( S ) = π .If equality in (7) holds, then we have equality in (8) for f = 1. Which meansthat λ = 0 and f = 1 is the first eigenfunction of L , i.e., Ric( N, N ) + | A | = 0. Onthe other hand, equality in (7) also implies equality in (10) and (11). Therefore,0 ≤ Z Σ ( R − dσ ≤ Z Σ ( R + | A | ) dσ −
12 Vol(Σ)= Z Σ R Σ dσ −
12 Vol(Σ) = 0 , thus Σ is totally geodesic and R = 12 on Σ. In particular, Ric( N, N ) = 0 on Σ.Also, from (6) we have R Σ = 12.To finish, observe that equality in (7) implies equality in (16). Therefore, since Y (Σ , [ g Σ ]) >
0, using Gursky’s theorem we obtain Y ( S , [ g can ]) ≥ Y (Σ , [ g Σ ]) ≥ (cid:18) π χ (Σ) − Z Σ | W Σ | dσ (cid:19) = 384 π = Y ( S , [ g can ]) . Then, Y (Σ , [ g Σ ]) = Y ( S , [ g can ]) and R Σ = 12, which imply by the solution of theYamabe problem and Obata’s theorem that Σ is isometric to S . (cid:3) ABRA˜AO MENDES
Remark 3.2.
It follows from the above proposition that if equality in (3) or (4)holds, then Σ is isometric to S up to scaling. In particular, Σ is Einstein. In thiscase, we can use Barros-Batista-Cruz-Sousa’s theorem to obtain Theorem 1.2 andTheorem 1.3. But, for the sake of completeness, we are going to present the proofsof these theorems here.Before proving our main results, we are going to state a very useful lemma dueto Bray, Brendle, and Neves [5] (see [12] for a more detailed proof). The sametechnique has been used by many authors in the literature (e.g. [1, 4, 6, 11]). Lemma 3.3.
Let M be a Riemannian -manifold. If Σ is a two-sided closedminimal hypersurface immersed in M such that Ric(
N, N ) + | A | = 0 on Σ , thenthere exists a smooth function w : ( − ε, ε ) × Σ → R , for some ε > , satisfying w (0 , x ) = 0 , ∂w∂t (0 , x ) = 1 and Z Σ ( w ( t, · ) − t ) dσ = 0 for all x ∈ Σ and t ∈ ( − ε, ε ) . Furthermore, Σ t = { exp x ( w ( t, x ) N ( x )) ∈ M : x ∈ Σ } is a closed hypersurface immersed in M with constant mean curvature for each t ∈ ( − ε, ε ) . Also, if Σ is embedded in M , then { Σ t } t ∈ ( − ε,ε ) is a foliation of aneighborhood of Σ = Σ . All entities associated to Σ t will be denoted with a subscript t , except the meancurvature which will be denoted by H ( t ). Furthermore, ρ t will denote the lapsefunction h ∂∂t , N t i . Theorem 3.4.
Let M be a Riemannian manifold with scalar curvature R sat-isfying inf M R > and nonnegative Ricci curvature. If Σ is a two-sided closedhypersurface embedded in M which is locally volume-minimizing, then the volumeof Σ satisfies Vol(Σ) (cid:18) inf M R (cid:19) ≤ Vol( S ) + 112 Z Σ | ˚Ric Σ | dσ. (17) Furthermore, if equality holds, then Σ is isometric to S and M is isometric to ( − ε, ε ) × S in a neighborhood of Σ , up to scaling.Proof. Inequality (17) follows immediately from Proposition 3.1, since all locallyvolume-minimizing hypersurfaces are stable minimal. Also, if equality in (17) holds,then Σ is isometric to S up to scaling and Ric( N, N ) = 0 = | A | on Σ. In particular,we can use Lemma 3.3. It is well known that dHdt ( t ) = − ∆ t ρ t − (Ric( N t , N t ) + | A t | ) ρ t . Since ρ = 1 and Σ is compact, we may assume that ρ t > t ∈ ( − ε, ε ).Therefore, using that Ric ≥ H ′ ( t ) is constant on Σ t , we have H ′ ( t ) Z Σ t ρ t dσ t = − Z Σ t ∆ t ρ t ρ t dσ t − Z Σ t (Ric( N t , N t ) + | A t | ) dσ t (18) ≤ − Z Σ t |∇ t ρ t | ρ t dσ t ≤ , (19) IGIDITY OF VOLUME-MINIMIZING HYPERSURFACES IN 5-MANIFOLDS 7 which imply H ′ ( t ) ≤ t ∈ ( − ε, ε ), and then H ( t ) ≤ ≤ H ( − t ) for all t ∈ [0 , ε ) , (20)because H (0) = 0. On the other hand, the first variation formula says that ddt Vol(Σ t ) = Z Σ t H ( t ) ρ t dσ t . (21)Then, (20) and (21) implyVol(Σ t ) ≤ Vol(Σ ) for all t ∈ ( − ε, ε ) . But, since Σ = Σ is locally volume-minimizing, we have Vol(Σ t ) = Vol(Σ) for all t ∈ ( − ε, ε ), for a smaller ε > ddt Vol(Σ t ) = Z Σ t H ( t ) ρ t dσ t and (20) imply H ( t ) = 0 for all t ∈ ( − ε, ε ). Using H ′ ( t ) = 0 into (18) and (19),we conclude that ρ t is constant on Σ t and Σ t is totally geodesic in M for each t ∈ ( − ε, ε ).Now, we want to prove that t N t ( x ) is a parallel vector field along to thecurve t G ( t, x ) = exp x ( w ( t, x ) N ( x )) for each x ∈ Σ. In fact, choosing a localcoordinate system x = ( x , x , x , x ) on Σ, we have (cid:28) ∇ ∂G∂t N t , ∂G∂x i (cid:29) = − (cid:28) N t , ∂ G∂x i ∂t (cid:29) = − ∂∂x i (cid:28) N t , ∂G∂t (cid:29) + (cid:28) ∇ ∂G∂xi N t , ∂G∂t (cid:29) = − ∂∂x i ρ t = 0 . Above we have used that ∇ ∂G∂xi N t = 0 since Σ t is totally geodesic. Also, D ∇ ∂G∂t N t , N t E = 12 ∂∂t h N t , N t i = 0 . Thus, N t is parallel.On the other hand, we know that t d (exp x ) w ( t,x ) N ( x ) N ( x ) is also parallelalong to t G ( t, x ). Then, N t ( x ) = ( d exp x ) w ( t,x ) N ( x ) N ( x ) by uniqueness ofparallel vector fields, since w (0 , x ) = 0 and N ( x ) = N ( x ). In particular, ρ t = (cid:28) ∂G∂t , N t (cid:29) = ∂w∂t . Now, because R Σ ( w ( t, · ) − t ) dσ = 0 and ρ t = ∂w∂t is constant on Σ t , we obtain0 = ddt Z Σ ( w ( t, · ) − t ) dσ = Z Σ (cid:18) ∂w∂t − (cid:19) dσ = (cid:18) ∂w∂t − (cid:19) Vol(Σ) , which imply ∂w∂t ( t, x ) = 1 for all ( t, x ) ∈ ( − ε, ε ) × Σ. Finally, because ∂w∂t (0 , x ) = 1,we have w ( t, x ) = t for all ( t, x ) ∈ ( − ε, ε ) × Σ. Therefore, G ( t, x ) = exp x ( tN ( x ))and we can easily check that G is an isometry from ( − ε, ε ) × Σ to a neighborhoodof Σ in M . (cid:3) Remark 3.5.
Supposing that Σ is immersed instead of embedded into M in theabove theorem, it follows from the same proof that Σ is isometric to S up to scalingand G is a local isometry from ( − ε, ε ) × Σ to M , if equality in (17) holds.The proof presented below is essentially the same as in [4], [5], and [12]. ABRA˜AO MENDES
Theorem 3.6.
Let M be a complete Riemannian manifold with scalar curvature R satisfying inf M R > and nonnegative Ricci curvature. Suppose that Σ is atwo-sided closed manifold embedded in M in such a way that Σ minimizes thevolume in its homotopy class. Then, the volume of Σ satisfies Vol(Σ) (cid:18) inf M R (cid:19) ≤ Vol( S ) + 112 Z Σ | ˚Ric Σ | dσ, (22) Moreover, if equality holds, then Σ is isometric to S and the Riemannian cover of M is isometric to R × S , up to scaling.Proof. Inequality (22) follows directly from Theorem 3.4. Suppose that equality in(22) holds and define G : R × Σ → M by G ( t, x ) = exp x ( tN ( x )). We claim that G is a local isometry. In fact, define I = { t > G | (0 ,t ) × Σ is a local isometry } and observe that Remark 3.5 implies Σ to be isometric to S up to scaling and I = ∅ . In particular, ˚Ric Σ = 0. On the other hand, it is not difficult to see that I is closed in (0 , ∞ ). In fact, suppose that t k ∈ I converges to t ∈ (0 , ∞ ). If t ≤ t k for some k then t ∈ I because (0 , t ) × Σ ⊂ (0 , t k ) × Σ and G | (0 ,t k ) × Σ is a localisometry. Otherwise, if t k < t for all k then S k (0 , t k ) × Σ = (0 , t ) × Σ implies that t ∈ I because G | (0 ,t k ) × Σ is a local isometry (which is a local property) for each k . Let us prove that I is also open. Given t ∈ I , we have that Σ t = G ( t, Σ) ishomotopic to Σ in M , Vol(Σ t ) = Vol(Σ), and ˚Ric Σ t = 0, because G : { t } × Σ → Σ t is a local isometric. In particular, Σ t minimizes the volume in its homotopy classand attains the equality in (22). Therefore, it follows from Remark 3.5 that thereexists ε > G | (0 ,t + ε ) × Σ is a local isometry. This proves that I is open in(0 , ∞ ). Thus, I = (0 , ∞ ), i.e., G | (0 , ∞ ) × Σ is a local isometry. Analogously, we canprove that G | ( −∞ , × Σ is a local isometry. This, together with Remark 3.5, impliesthat G is a local isometry. In particular, G is a covering map. (cid:3) References
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