Mean curvature in manifolds with Ricci curvature bounded from below
aa r X i v : . [ m a t h . DG ] M a y MEAN CURVATURE IN MANIFOLDS WITH RICCI CURVATUREBOUNDED FROM BELOW
JAIGYOUNG CHOE AND AILANA FRASER
Abstract.
Let M be a compact Riemannian manifold of nonnegative Ricci curvature andΣ a compact embedded 2-sided minimal hypersurface in M . It is proved that there is adichotomy: If Σ does not separate M then Σ is totally geodesic and M \ Σ is isometric tothe Riemannian product Σ × ( a, b ), and if Σ separates M then the map i ∗ : π (Σ) → π ( M )induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sidedhypersurface with mean curvature H ≥ ( n − √ k in a manifold of Ricci curvature Ric M ≥− ( n − k, k >
0, and for a free boundary minimal hypersurface in a manifold of nonnegativeRicci curvature with nonempty strictly convex boundary. As an application it is shown thata compact n -dimensional manifold N with the number of generators of π ( N ) < n cannotbe minimally embedded in the flat torus T n +1 . introduction Euclid’s fifth postulate implies that there exist two nonintersecting lines on a plane. Butthe same is not true on a sphere, a non-Euclidean plane. Hadamard [Ha] generalized this toprove that every geodesic must meet every closed geodesic on a surface of positive curvature.Note that an n -dimensional minimal submanifold of a Riemannian manifold M k is a criticalset for n -dimensional area. Replacing the geodesic with the minimal submanifold, Frankel [F]further generalized Hadamard’s theorem: Let Σ and Σ be immersed minimal hypersurfacesin a complete connected Riemannian manifold M n +1 of positive Ricci curvature. If Σ is compact, then Σ and Σ must intersect. It should be remarked that a manifold ofnonnegative Ricci curvature like S × S has many disjoint minimal spheres.Using the connectivity of the inverse image of Σ under the projection map in the universalcover of M , Frankel also proved that the natural homomorphism of fundamental groups: π (Σ ) → π ( M ) is surjective. This means that the minimality of Σ imposes restrictions on π (Σ ). This reminds us of a similar restriction on π ( M ), as proved by Myers [M], that if M has positive Ricci curvature, then π ( M ) is finite.These two theorems of Frankel have the dual versions in the negatively curved case asfollows: If M n is a complete Riemannian manifold of nonpositive sectional curvature, thenevery compact immersed minimal submanifold Σ k must have an infinite fundamental groupand moreover, if Σ k is totally geodesic, then π (Σ k ) → π ( M n ) is 1-1 [He].It was Lawson [L] who first realized the topological implication of Frankel’s theorem; hefound that Frankel’s proof of the surjectivity works also for each component of M \ Σ if M is a compact connected orientable Riemannian manifold of positive Ricci curvature and Σis a compact embedded orientable minimal hypersurface. He then showed that π ( ¯ D j , Σ) =0 , j = 1 ,
2, where M \ Σ = D ∪ D . This implies that Σ has as many 1-dimensional holes J.C. supported in part by NRF 2011-0030044, SRC-GAIA, A.F. supported in part by NSERC. (loops) as D j does. Hence D and D are handlebodies and since M is diffeomorphic to S ,Σ is unknotted.Recently Petersen and Wilhelm [PW] gave a new proof of Frankel’s generalized Hadamardtheorem. They also showed that if M has nonnegative Ricci curvature and has two nonin-tersecting minimal hypersurfaces, then these are totally geodesic and a rigidity phenomenonoccurs. Whereas Frankel and Lawson used the second variation formula for arc length, Pe-tersen and Wilhelm utilized the superhamonicity of the distance function from a minimalhypersurface. It should be mentioned that Cheeger and Gromoll had used the superhar-monicity of the distance function arising from a minimizing geodesic [CG]. See also [CK],[Wy].In this paper we show that there is a dichotomy for a compact Riemannian manifold ofnonnegative Ricci curvature (Theorem 2.5): A compact embedded 2-sided minimal hyper-surface Σ does not separate M or separates M into two nonempty components D and D ,and consequently, Σ is totally geodesic and M is isometric to a mapping torus or the map i ∗ : π (Σ) → π ( ¯ D j ) , j = 1 , , induced by inclusion is surjective. As a result M cannot havemore 1-dimensional holes than Σ unless M is diffeomorphic to Σ × S . The first part ofTheorem 2.5 reminds us of the Cheeger-Gromoll splitting theorem [CG]; their line is dual toour nonseparating minimal hypersurface. See also [CaGa].The surjectivity of i ∗ : π (Σ) → π ( ¯ D j ) is obtained in more general settings as follows.Let M n be a Riemannian manifold of Ricci curvature Ric M ≥ − ( n − k, k > M . If Ω is meanconvex with H (Σ) ≥ ( n − √ k , then Σ is connected and i ∗ : π (Σ) → π ( ¯Ω i ) is surjective.Thus if n = 3 then Ω is a handlebody.We also consider the case when the compact Riemannian manifold M n of nonnegative Riccicurvature has nonempty boundary ∂M which is strictly convex with respect to the inwardunit normal. Fraser and Li [FL] showed that any two properly embedded orientable minimalhypersurfaces in M meeting ∂M orthogonally must intersect. They also showed that if Σ isa properly embedded orientable minimal hypersurface in M meeting ∂M orthogonally, thenΣ divides M into two connected components D and D . Generalizing [FL], we show thatthe maps i ∗ : π (Σ) → π ( ¯ M ) and i ∗ : π (Σ) → π ( ¯ D j ), j = 1 ,
2, are surjective and bothcomponents of M \ Σ are handlebodies. When n = 3 it is shown that Σ is unknotted. Wealso prove some corresponding results in the case where Ric M ≥ − ( n − k, k > N be an n -dimensional compact manifold with the number of generators of π ( N ) = k that is minimally embedded in the flat ( n + 1)-torus T n +1 . Then we must have k ≥ n . If k = n , then N ≈ T n , and if k > n , then T n +1 \ N has two components D , D such that the number of generators of π ( D j ) is bigger than n, j = 1 ,
2. This is a higherdimensional generalization of Meeks’ theorem [Mk] that a compact surface of genus 2 cannotbe minimally immersed in T . 2. surjectivity It is well known that the second variation of arc length involves negative the integral ofthe sectional curvature. It is for this reason that the Ricci curvature affects both the meancurvature of the level surfaces of the distance function and the Laplacian of the distancefunction. The following lemma verifies this influence.
EAN CURVATURE IN MANIFOLDS WITH RICCI CURVATURE BOUNDED FROM BELOW 3
Lemma 2.1.
Assume that M n +1 is a complete Riemannian manifold of nonnegative Riccicurvature. Let D be a domain in M and N ⊂ ∂D a hypersurface with mean curvature H N ≥ c with respect to the inward unit normal ν to N , i.e., H N = h ~H N , ν i . Suppose that thedistance function d from N is well defined in D . Then at a point q ∈ D where d is smooth a) the level surface of d through q has mean curvature ≥ c with respect to the unit normalaway from N ( in fact, that mean curvature is monotone nondecreasing in d ) ; b) ∆ d ≤ − c ;c) the level surfaces of d are piecewise smooth, and at a nonsmooth point where two smoothlevel surfaces intersect, they make an angle < π in the direction away from N .Proof. Let S be a smooth level surface of d through a point q ∈ D . Let γ ⊂ D be thegeodesic up to q that realizes the distance from N and is parametrized by arc length. Then γ hits S and N orthogonally at q and at a point p ∈ N . Choose any unit vector v tangentto N at p and parallel translate it along γ to q , obtaining a unit parallel vector field V along γ which is normal to γ and tangent to S at q . Consider the lengths of the curves obtainedby moving γ in the direction of V . Then the second variation formula and the assumptionthat S is a level surface of d give us L ′′ V = II S ( V, V ) − II N ( V, V ) − Z γ K ( V, γ ′ ) ≥ , where II denotes the second fundamental form defined by II ( u, v ) = h∇ u v, ν i with respectto the inward unit normal ν away from N , and K ( V, γ ′ ) is the sectional curvature on thespan of V and γ ′ . We can compute the same for orthonormal vectors v , . . . , v n spanning thetangent space to N at p and sum up the above inequalities for the corresponding orthonormalparallel vector fields V , . . . , V n , to get H S ( q ) − H N ( p ) − Z γ Ric( γ ′ , γ ′ ) ≥ , which proves a) because R γ Ric( γ ′ , γ ′ ) is monotone nondecreasing in d . Let E , . . . , E n beorthonormal vector fields on S in a neighborhood of q . Extend them to orthonormal vectorfields ¯ E , . . . , ¯ E n , ¯ E n +1 on M in a neighborhood of q such that ¯ E n +1 = γ ′ . Then at q ∆ d = n +1 X i =1 (cid:2) ¯ E i ¯ E i ( d ) − ( ∇ ¯ E i ¯ E i ) d (cid:3) = − H S ( q ) ≤ − H N ( p ) ≤ − c. This proves b).To prove c) suppose there are two distinct points p and p in N with dist( p i , q ) = d ( q ).Let N i ⊂ N be a small neighborhood of p i , i = 1 ,
2. Then d i ( · ) := dist( · , N i ) is a welldefined function which is smooth in a neighborhood of q . Furthermore, it is easy to see that d ( · ) = min { d ( · ) , d ( · ) } . Therefore in a neighborhood of qd − { r ≥ d ( q ) } = d − { r ≥ d ( q ) } ∩ d − { r ≥ d ( q ) } . The proof is complete. (cid:3)
It follows from Lemma 2.1 that the distance function from a minimal hypersurface in amanifold of nonnegative Ricci curvature is superharmonic at points where it is smooth. Inthe following lemma we show that the distance function is superharmonic in the barrier
J. CHOE AND A. FRASER sense at points where it is not smooth, and hence satisfies the maximum principle ([C], [Pe]Theorem 66), that is, it is constant in a neighborhood of every local minimum.
Lemma 2.2.
Let Σ be a minimal hypersurface in a complete Riemannian manifold M ofnonnegative Ricci curvature. Then the distance function d from Σ is superharmonic ∆ d ≤ in the barrier sense. That is, given p ∈ M , for every ε > there exists a smooth supportfunction from above d ε defined in a neighborhood of p such that: (1) d ε ( p ) = d ( p ) , (2) d ( x ) ≤ d ε ( x ) in some neighborhood of p , (3) ∆ d ε ( p ) ≤ ε .Proof. By Lemma 2.1 b) we know that ∆ d ≤ d is smooth. For any other p ∈ M choose a unit speed minimizing geodesic γ : [0 , l ] → M between Σ and p , with γ (0) ∈ Σand γ ( l ) = p . Let ν be the unit normal of Σ near γ (0) in the direction toward p , let ϕ ( t ) = e − t / (1 − t ) be a smooth cut-off function, and defineΣ δ = { exp x δϕ ( d Σ ( γ (0) , x )) ν ( x ) : x ∈ Σ ∩ B r ( γ (0)) } for small δ > r >
0. Since Σ δ is a small perturbation of Σ, we have | H Σ δ | ≤ C ( δ ) with C ( δ ) → δ →
0. Given ε >
0, choose δ = δ ( ε ) sufficiently small so that C ( δ ) ≤ ε . Weclaim that d ε ( · ) := δ ( ε ) + d (Σ δ ( ε ) , · ) is a smooth support function from above for d at p . Itis clear from the construction that d ε ( p ) = d ( p ). If x is sufficiently close to p , there is aninterior point x ′ in Σ δ that realizes the distance from x to Σ δ . By the construction of Σ δ , d (Σ , x ′ ) ≤ δ , and we have d ( x ) = d (Σ , x ) ≤ d (Σ , x ′ ) + d ( x ′ , x ) ≤ δ + d (Σ δ , x ) = d ε ( x ) . If d ε is smooth at p , then by Lemma 2.1 b), ∆ d ε ( p ) ≤ C ( δ ) ≤ ε . It remains to showsmoothness. Suppose d ε is not smooth at p . Then we know that either(1) there are two minimizing geodesics from p to Σ δ , or(2) p is a focal point of Σ δ .In case (1), there is a minimizing geodesic from p to a point q = γ ( δ ) in Σ δ . But byconstruction of Σ δ , d (Σ , q ) < δ , and so this implies that d (Σ , p ) ≤ d (Σ , q ) + d ( q, p ) < δ + d (Σ δ , p ) = l, a contradiction. In case (2), if p is a focal point of Σ δ , there is a Jacobi field J along γ | [ δ,l ] with J ( δ ) tangent to Σ δ at γ ( δ ), J ( l ) = 0, and such that J ′ ( δ ) + S γ ′ ( δ ) ( J ( δ )) is orthogonalto Σ δ , where S γ ′ ( δ ) is the linear operator on T γ ( δ ) Σ δ given by the second fundamental formof Σ δ in M , that is, S γ ′ ( δ ) X = − ( ∇ X γ ′ ( δ )) T , X ∈ T γ ( δ ) Σ δ . The second variation of lengthof γ | [ δ,l ] in the direction J is zero: I ( J, J ) = Z lδ (cid:2) |∇ γ ′ J | − h R ( J, γ ′ ) γ ′ , J i (cid:3) dt + h∇ J J, γ ′ i (cid:12)(cid:12)(cid:12) lδ = − Z lδ h J ′′ + R ( J, γ ′ ) γ ′ , J i dt + [ h∇ γ ′ J, J i + h∇ J J, γ ′ i ] (cid:12)(cid:12)(cid:12) lδ = − Z lδ h J ′′ + R ( J, γ ′ ) γ ′ , J i dt − h J ′ ( δ ) + S γ ′ ( δ ) ( J ( δ )) , J ( δ ) i = 0 . EAN CURVATURE IN MANIFOLDS WITH RICCI CURVATURE BOUNDED FROM BELOW 5
Let σ be the geodesic in Σ δ with σ (0) = γ ( δ ) and σ ′ (0) = J ( δ ). For δ small, there is aunique minimizing geodesic γ s between σ ( s ) and Σ, and since γ ( δ ) is the point on Σ δ thatis furthest from Σ, d ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 L ( γ s ) < . Let W be the variation field of the variation γ s of γ | [0 ,δ ] . Then W ( δ ) = J ( δ ), and for thevector field V along γ given by V ( t ) = ( W ( t ) for 0 ≤ t ≤ δJ ( t ) for δ ≤ t ≤ l, the second variation of length of γ is strictly less than zero. This contradicts the fact that γ is a minimizing geodesic from p to Σ. Therefore, d ε is smooth in a neighborhood of p and isa smooth support function from above for d at p . (cid:3) With the superharmonicity of the distance function in our hands we are now able to provethe main theorem.
Definition 2.3.
Let Σ be a compact connected embedded hypersurface in a compact n -manifold M . Σ is said to be separating if M \ Σ has two nonempty connected components,and nonseparating if M \ Σ is connected.
Definition 2.4. A handlebody is a 3-manifold with boundary which is homeomorphic to aclosed regular neighborhood of a connected properly embedded 1-dimensional CW complexin R . A surface Σ in a 3-manifold M is called a Heegaard surface if Σ separates M into twohandlebodies. Theorem 2.5.
Let M be a compact Riemannian n-manifold of nonnegative Ricci curvatureand Σ a compact connected embedded 2-sided minimal hypersurface in M . Then either a) Σ is nonseparating and totally geodesic and M is isometric to a mapping torus Σ × [0 , a ]( x, ∼ ( y, a ) iff φ ( x ) = y , where φ : Σ → Σ is an isometry, or b) Σ is separating, and if D , D ⊂ M are the components of M \ Σ , then for j = 1 , the maps i ∗ : π (Σ) → π ( ¯ D j ) , i ∗ : π (Σ) → π ( M ) and i ∗ : π ( D j ) → π ( M ) induced by the inclusion are all surjective.If n = 3 and Σ is a separating, then Σ is a Heegaard surface.Proof. a) Choose a function that is equal to 0 on Σ and in a neighborhood of one side ofΣ and equal to 1 in a neighborhood of the other side of Σ. Since Σ is nonseparating, thisfunction can be extended to a smooth function on M \ Σ, and by passing to the quotientmod Z we obtain a nonconstant smooth function f : M → R / Z = S . Let f ∗ : π ( M ) → Z be the induced map on the fundamental groups, and for the universalcover ˜ M of M , consider the cyclic cover ˆ M = ˜ M /G of M corresponding to the subgroup J. CHOE AND A. FRASER G = ker f ∗ of π ( M ). Since ˆ M has a geodesic line, the result follows from the splittingtheorem [CG]. However, we will give an alternate direct proof, which will be needed for theproof of part b).Let Σ , Σ ⊂ ˆ M be two adjacent preimages of Σ under the projection π : ˆ M → M suchthat Σ and Σ bound a connected domain D ⊂ ˆ M on which π is 1-1. Here we adopt thearguments of [PW]. If d i is the distance function on D to Σ i , then our hypotheses on theRicci curvature of M and the minimality of Σ i imply that ∆ d i ≤ d + d is also superharmonic in the barrier sense. But it has an interiorminimum on a minimal geodesic γ between Σ and Σ and so by the maximum principle itis constant on D . Then it follows that d i is harmonic and smooth on D . Recall the Bochnerformula for a smooth function u on ˆ M :12 ∆ | du | = | Hess u | + h∇ u, ∇ (∆ u ) i + Ric( ∇ u, ∇ u ) . Since | du | = 1 for u = d i , the formula yields Hess d i = 0 on D . Therefore Σ i is totallygeodesic and ˆ M is isometric to Σ i × R . Thus M is isometric to a mapping torusΣ × [0 , a ]( x, ∼ ( y, a ) iff φ ( x ) = y , where a is the length of γ and φ : Σ → Σ is an isometry.b) Let π : ˜ D j → D j be the universal cover of D j , j = 1 ,
2. Extending π up to ∂ ˜ D j , weclaim that ∂ ˜ D j = π − (Σ) is connected. If ∂ ˜ D j is not connected, let d = inf { d (Σ ′ , Σ ′′ ) : Σ ′ and Σ ′′ are distinct components of ∂ ˜ D j } . As in [L], there exist components Σ ′ and Σ ′′ such that there is a geodesic γ in ˜ D j from Σ ′ to Σ ′′ of length d . By continuity, there is a neighborhood of γ in ˜ D j such that the distancefunctions d ′ and d ′′ to Σ ′ and to Σ ′′ in ˜ D j are well defined. By the same arguments as ina) we see that d ′ + d ′′ is superharmonic. Note that d ′ + d ′′ has interior minimum at allpoints of γ . As in a), it follows that a neighborhood of γ is isometric to a product manifold(Σ ′ ∩ U ) × (0 , d ), where U is a neighborhood of γ (0) in Σ ′ . Let U be the set of points inΣ ′ that can be connected to Σ ′′ by a geodesic of length d . By the argument above, U isopen and U ⊂ U . We claim that U is also closed. To see this, let p m be a sequence of pointsin U converging to p ∈ Σ ′ , and let γ m be a geodesic in ˜ D j of length d from p m to Σ ′′ . Bypassing to a subsequence we can see that there exists a geodesic γ of length d from p toΣ ′′ such that { γ m } converges to γ . It may happen that γ hits ∂ ˜ D j \ Σ ′ at some point q with dist( p, q ) < d . But then dist( p, ∂ ˜ D j \ Σ ′ ) < d , which is a contradiction. Therefore p ∈ U and U is closed. Since U is both open and closed, U = Σ ′ . Therefore ˜ D j is isometricto the product manifold Σ ′ × (0 , d ). Hence M \ Σ is isometric to Σ × (0 , d ), and so M is diffeomorphic (not necessarily isometric) to Σ × S . But then Σ is nonseparating in M ,which is a contradiction. Hence ∂ ˜ D j is connected, as claimed.Let ℓ be a loop in ¯ D j with base point p ∈ Σ. Lift ℓ to a curve ˜ ℓ in ˜ D j from p ∈ π − ( p )to p ∈ π − ( p ). Since π − (Σ) is connected, there is a curve ˆ ℓ in π − (Σ) connecting p to p .Moreover, ˆ ℓ is homotopic to ˜ ℓ in ˜ D j as ˜ D j is simply connected. Hence π (ˆ ℓ ) is a loop in Σthat is homotopic in D j to ℓ . Therefore the map i ∗ : π (Σ) → π ( ¯ D j ) induced by inclusionis surjective. EAN CURVATURE IN MANIFOLDS WITH RICCI CURVATURE BOUNDED FROM BELOW 7
Let ℓ be a loop in M . Divide ℓ into two parts ℓ , ℓ such that ℓ j ⊂ D j . Cover ℓ j with acurve ˜ ℓ j in ˜ D j as above. By the connectedness of π − (Σ) again we have a curve ˆ ℓ j in π − (Σ)with the same end points as ˜ ℓ j and homotopic to ˜ ℓ j . Then π (ˆ ℓ ) ∪ π (ˆ ℓ ) is a loop in Σ thatis homotopic to ℓ . Hence i ∗ : π (Σ) → π ( M ) is also surjective.Since Σ is 2-sided there exists ¯ ℓ j ⊂ D j which is very close to π (ˆ ℓ ) ∪ π (ˆ ℓ ). Hence ¯ ℓ j and π (ˆ ℓ ) ∪ π (ˆ ℓ ) are homotopic and therefore i ∗ : π ( D j ) → π ( M ) is surjective.To prove the final statement, suppose n = 3. The surjectivity of i ∗ : π (Σ) → π ( ¯ D j )implies π ( ¯ D j , Σ) ≈
0. Then by [Pa] we can use the Loop Theorem and Dehn’s Lemma toshow that ¯ D and ¯ D are handlebodies. Thus Σ is a Heegaard surface. (cid:3) Corollary 2.6.
Let Σ be a compact connected embedded minimal surface in a Riemannianthree-sphere M of nonnegative Ricci curvature. Then Σ is unknotted.Proof. From the Jordan-Brouwer separation theorem it follows that Σ is separating. We showthat Σ is unknotted in the sense that if Σ ′ is a standardly embedded surface of the samegenus as Σ in M , then there exists an orientation preserving diffeomorphism f : M → M such that f (Σ) = Σ ′ . By Theorem 2.5 b), Σ is a Heegaard surface. It follows from [W] thatthere is a PL homeomorphism ˜ f : M → M such that ˜ f (Σ) = Σ ′ . Then by results from [HM]there exists a smooth map f as claimed. (cid:3) Remark . It should be mentioned that Meeks and Rosenberg [MR] showed a noncompactproperly embedded minimal surface in S × R is unknotted.The result of Frankel shows that two compact minimal hypersurfaces in a manifold ofpositive Ricci curvature must intersect. However, a manifold of nonnegative Ricci curvaturecan have many disjoint compact minimal hypersurfaces. Furthermore, in the case of negativecurvature, there can even exist disjoint hypersurfaces that bound a mean convex region;for example, spheres equidistant to two disjoint planes in hyperbolic space. On the otherhand two disjoint horospheres in hyperbolic space cannot bound a mean convex region.This suggests that there can only exist a mean convex region with two disjoint boundarycomponents if the mean curvature is less than a critical number involving a lower bound onthe curvature of the ambient manifold.Here we show this, that in fact Frankel’s argument can be extended to the case of manifoldsof negative Ricci curvature provided the Ricci curvature is bounded from below and thehypersurfaces have mean curvature that is sufficiently large. We obtain the correspondingresult on surjectivity of the natural homomorphism of fundamental groups for compact 2-sided hypersurfaces with mean curvature above this critical (sharp) threshold involving thelower bound on the Ricci curvature. In the 3-dimensional case, such hypersurfaces mustbound handlebodies; for example, a compact connected 2-sided hypersurface with meancurvature | H | ≥ Theorem 2.8.
Let M n be a complete Riemannian manifold of Ricci curvature bounded frombelow, Ric M ≥ − ( n − k , k > . Let Σ be a compact hypersurface that bounds a connectedregion Ω in M . Suppose that the mean curvature vector of Σ points everywhere into Ω , and H ≥ ( n − √ k . Then Σ is connected, and the map i ∗ : π (Σ) → π ( ¯Ω) induced by the inclusion is surjective. If n = 3 then Ω is a handlebody. J. CHOE AND A. FRASER
Proof.
We argue by contradiction. Suppose Σ is not connected. Let Σ and Σ be distinctconnected components of Σ. Then there exists a unit speed geodesic γ : [ − l/ , l/ → M with γ ( − l/
2) = p ∈ Σ and γ ( l/
2) = q ∈ Σ that realizes distance from Σ to Σ , andmeets Σ orthogonally at the endpoints on the mean convex side of Σ. Let e , . . . , e n − bean orthonormal basis for the tangent space to Σ at p , and parallel transport to obtainparallel orthonormal vector fields E , . . . , E n − along γ . Since γ meets Σ orthogonally, E ( q ) , . . . , E n − ( q ) are tangent to Σ at q . Let V i ( t ) = ϕ ( t ) E i ( t ) with ϕ ( t ) = c ( l ) cosh( √ k t )and c ( l ) = cosh( √ k l/ ϕ ′′ − kϕ = 0 and ϕ ( − l/
2) = ϕ ( l/
2) = 1. Consider thesum of the second variations of length of γ in the directions V i :0 ≤ n − X i =1 L ′′ V i (0)= Z l/ − l/ (cid:2) ( n − ϕ ′ ) − ϕ Ric( γ ′ γ ′ ) (cid:3) dt + n − X i =1 ϕ h∇ E i E i , γ ′ i (cid:12)(cid:12)(cid:12) l/ − l/ = − Z l/ − l/ (cid:2) ( n − ϕϕ ′′ + ϕ Ric( γ ′ γ ′ ) (cid:3) dt − H Σ ( p ) − H Σ ( q ) + ( n − ϕϕ ′ (cid:12)(cid:12)(cid:12) l/ − l/ ≤ − ( n − Z l/ l/ ϕ ( ϕ ′′ − kϕ ) dt − n − √ k + 2( n − √ k tanh( √ k l/ − n − √ k + 2( n − √ k tanh( √ k l/ < , which is a contradiction. Therefore Σ is connected. Similarly π − (Σ) is connected in theuniversal cover ˜Ω of Ω under the covering map π : ˜Ω → Ω. It then follows by arguments asin the proofs of Theorem 2.5 b) that i ∗ : π (Σ) → π ( ¯Ω) is surjective, and if n = 3 then Ω isa handlebody. (cid:3) Remark . The assumption that Σ bounds a region is not necessary. If M n is a completeRiemannian manifold with Ric M ≥ − ( n − k , k >
0, and Σ is a compact 2-sided hypersurfacein M with | H | ≥ ( n − √ k , then it follows that Σ bounds a collection of disjoint connectedregions Ω , . . . , Ω s in M such that the mean curvature vector ~H points everywhere into Ω i ,and each has as boundary ∂ Ω i a connected component of Σ. To see this, first observe thateach component Σ ′ of Σ is separating. If not, we may construct a cyclic cover ˆ M of M asin the proof of Theorem 2.5 a). Then Ric ˆ M ≥ − ( n − k , and each component of π − (Σ ′ )divides ˆ M into two infinite pieces. For one of these pieces, Ω, the mean curvature vector ~H of ∂ Ω points everywhere into Ω and satisfies | H | ≥ ( n − √ k . It follows from [S] Lemma 1that Vol(Ω) ≤ n − ∂ Ω) < ∞ , a contradiction. Therefore each component of Σ is separating, and hence Σ bounds a col-lection of disjoint regions Ω , . . . , Ω s such that ~H points everywhere into Ω i for i = 1 , . . . , s .Finally, Theorem 2.8 implies that ∂ Ω i is connected for each i and hence is a connectedcomponent of Σ. EAN CURVATURE IN MANIFOLDS WITH RICCI CURVATURE BOUNDED FROM BELOW 9
Remark . This theorem is sharp in the sense that on a hyperbolic surface, disjointcircles of curvature 1 cannot bound a convex region, but on a hyperbolic surface with acusp there exists a convex annular region with two boundary components (cross sectionsof the cusp) that have curvature slightly less than, but arbitrarily close to 1. One canconstruct analogous compact examples in higher dimensions in quotients of hyperbolic space H n . Disjoint horospheres with H = n − n -space H n , but in the half-space model of H n there can exist a convex region bounded bythe two hyperplanes P , P with ∂P = ∂P ⊂ ∂ H n and making angles θ, π − θ with ∂ H n ; theboundary components have mean curvature slightly less than, but arbitrarily close to n − θ →
0. 3. convex domain
In this section the Riemannian manifold M n will be assumed to have nonempty boundary ∂M . Suppose that M has nonnegative Ricci curvature and ∂M is strictly convex. Recallthat Frankel [F] showed that two compact immersed minimal hypersurfaces in a Riemannianmanifold M of positive Ricci curvature must intersect. Fraser and Li ([FL], Lemma 2.4)extended Frankel’s theorem to two properly embedded minimal hypersurfaces Σ , Σ in M ,i.e., ∂ Σ i ⊂ ∂M, i = 1 ,
2, meeting ∂M orthogonally. They also showed ([FL] Corollary2.10) that if Σ is a properly embedded orientable minimal hypersurface in M meeting ∂M orthogonally, then Σ divides M into two connected components D and D . We show thatthe maps i ∗ : π (Σ) → π ( ¯ D j ), j = 1 ,
2, are surjective and that Σ is unknotted when n = 3.We also prove some corresponding results in the case where the Ricci curvature is boundedfrom below by a negative constant. Lemma 3.1.
Let M be an n -dimensional compact manifold of nonnegative Ricci curva-ture with strictly convex boundary ∂M . Suppose that Σ is a properly embedded minimalhypersurface in M meeting ∂M orthogonally. Then the maps i ∗ : π (Σ) → π ( M ) and i ∗ : π (Σ) → π ( ¯ D j ) , j = 1 , , are surjective, where D , D are the components of M \ Σ .Proof. Let ˜ D j be the universal cover of D j with the projection map π : ˜ D j → D j . Since π − (Σ) is connected, by the same arguments as in the proof of Theorem 2.5 b) we easily getthe surjectivity of i ∗ : π (Σ) → π ( ¯ D j ). Applying the same arguments to π : ˜ M → M , weget the surjectivity of i ∗ : π (Σ) → π ( M ) as well. (cid:3) Theorem 3.2.
Let M be a -dimensional compact orientable Riemannian manifold of non-negative Ricci curvature. Suppose M has nonempty boundary ∂M which is strictly convexwith respect to the inward unit normal. Then an orientable properly embedded minimal sur-face Σ in M meeting ∂M orthogonally divides M into two handlebodies.Proof. By Lemma 3.1 we have π ( ¯ D , Σ) = π ( ¯ D , Σ) = 0. As in the proof of Theorem 2.5,using the Loop Theorem and Dehn’s Lemma, we conclude that D and D are handlebodies. (cid:3) Corollary 3.3.
Let M be a 3-dimensional compact Riemannian manifold of nonnegativeRicci curvature with nonempty strictly convex boundary ∂M . Then any orientable properlyembedded minimal surface Σ in M orthogonal to ∂M is unknotted. Proof. M is diffeomorphic to the 3-ball B by Theorem 2.11, [FL]. Σ divides M into twohandlebodies by Theorem 3.2. Let ˇ M be the doubling of M , and let ˇΣ be the doubling of Σin M . Then ˇ M is diffeomorphic to S . By [W] and [HM] as in Corollary 2.6, ˇΣ is unknottedin ˇ M and Σ is unknotted in M . (cid:3) We also have a version in the case of curvature with a negative lower bound.
Theorem 3.4.
Let M n be a compact Riemannian manifold with nonempty boundary. Sup-pose M has Ricci curvature bounded from below Ric M ≥ − ( n − k , k > , and the boundary ∂M is strictly convex with respect to the inward unit normal. Let Σ be a hypersurface in M that bounds a connected region Ω in M and makes a constant contact angle θ ≤ π/ with ∂ Ω ∩ ∂M . Suppose that the mean curvature vector of Σ points everywhere into Ω , and H ≥ ( n − √ k . Then Σ is connected, and the map i ∗ : π (Σ) → π ( ¯Ω) induced by the inclusion is surjective.Proof. Suppose Σ is not connected. Let Σ and Σ be two distinct connected componentsof Σ. Let d and d be the distance functions on Ω from Σ and Σ respectively. Since ∂M is convex and Σ i , i = 1 ,
2, makes a contact angle ≤ π/ ∂ Ω ∩ ∂M , for any point x inΩ \ Σ i , d i ( x ) is realized by a geodesic in Ω from x to an interior point y on Σ i . Then thereexists a geodesic γ in Ω from some interior point p ∈ Σ to some interior point q ∈ Σ , thatrealizes the distance from Σ to Σ , and meets Σ and Σ orthogonally. But as in the proofof Theorem 2.8 the Ricci curvature lower bound and assumption on the mean curvature ofΣ and Σ imply that γ is unstable, a contradiction. Therefore Σ is connected.Let ˜Ω be the universal cover of Ω with projection map π : ˜Ω → Ω. By the same argumentas above, ∂ ˜Ω \ π − ( ∂M ) must be connected, and we easily get the surjectivity of i ∗ : π (Σ) → π ( ¯Ω). (cid:3) Corollary 3.5.
Under the assumptions of Theorem 3.4, if n = 3 then Ω is a handlebody. nonexistence As an application of the surjectivity of i ∗ : π (Σ) → π ( M ) Frankel showed that S n cannotbe minimally embedded in P n +1 . In this section we further utilize the surjectivity of i ∗ andprove nonexistence of some minimal embeddings in T n +1 .Meeks [Mk] proved that a compact surface of genus 2 cannot be minimally immersed in aflat 3-torus T . He used the fact that the Gauss map of a minimal surface Σ ⊂ T into S has degree one. A theorem of a similar nature can be proved in higher dimension by usingthe surjectivity of i ∗ . Theorem 4.1.
Let N be a compact orientable n -dimensional manifold with the number ofgenerators of π ( N ) = k and let T n +1 be the ( n + 1) -dimensional flat torus. a) If k < n , N cannot be minimally embedded in T n +1 . b) If k = n and N is minimally embedded in T n +1 , then N is a flat T n . c) If k > n and N is minimally embedded in T n +1 , then N is separating and the number ofgenerators of π ( D j ) must be bigger than n for j = 1 , D ∪ D = T n +1 \ N ) . EAN CURVATURE IN MANIFOLDS WITH RICCI CURVATURE BOUNDED FROM BELOW 11
Proof.
Let N n be an embedded minimal submanifold in T n +1 with k ≤ n . Then the map i ∗ : π ( N ) → π ( T n +1 ) is not surjective. Hence from Theorem 2.5 we conclude that N isnonseparating and totally geodesic in T n +1 . Hence N is a flat T n and k = n . Therefore N cannot be minimally embedded in T n +1 in case k < n . If k > n , then N must be separatingand c) follows from the surjectivity of i ∗ : π ( D j ) → π ( T n +1 ) in Theorem 2.5 b). (cid:3) Remark . In case n = 2, Theorem 4.1 c) gives a new proof of the Meeks theorem mentionedabove.Let Γ k ⊂ R be the union of k loops γ , . . . , γ k in R with γ i ∩ γ j = { p } for every pair1 ≤ i, j ≤ k and let Γ n +1 k be the ε -tubular neighborhood of Γ k in R n +1 . Γ n +1 k can be seen asa high-dimensional handlebody in R n +1 . Note that ∂ Γ n +1 k is diffeomorphic to k ( S n − × S ),the connected sum of k copies of S n − × S , and that π ( ∂ Γ n +1 k ) has k generators when n ≥ ∂ Γ n +1 n is not diffeomorphic to T n , Theorem 4.1 implies that ∂ Γ n +1 k cannot be minimallyembedded in T n +1 for any k = 1 , . . . , n .Schwarz’s P -surface is a minimal surface of genus 3 in the cubic torus T . One cangeneralize this surface to higher dimension as follows. T n +1 has a 1-dimensional skeleton L n +1 which is homeomorphic to Γ n +1 . There also exists its dual L ′ n +1 that is a paralleltranslation of L n +1 . One can foliate T n +1 \ ( L n +1 ∪ L ′ n +1 ) by 1-parameter family of n -dimensional hypersurfaces which are diffeomorphic to ∂ Γ n +1 n +1 and sweeping out from L n +1 to L ′ n +1 . Applying the minimax argument, one could find a minimal hypersurface Σ from thisfamily of hypersurfaces [CH]. Σ should be diffeomorphic to ∂ Γ n +1 n +1 and π (Σ) should have n + 1 generators. Therefore the upper bound n in Theorem 4.1 is sharp. References [C] E. Calabi,
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Jaigyoung Choe: Korea Institute for Advanced Study, Seoul, 02455, Korea
E-mail address : [email protected] Ailana Fraser: Department of Mathematics, University of British Columbia, 121-1984Mathematics Road, Vancouver, BC V6T 1Z2, Canada
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