On free boundary minimal hypersurfaces in the Riemannian Schwarzschild space
aa r X i v : . [ m a t h . DG ] F e b ON FREE BOUNDARY MINIMAL HYPERSURFACES IN THERIEMANNIAN SCHWARZSCHILD SPACE
EZEQUIEL BARBOSA AND JOS´E M. ESPINAR
Abstract.
In contrast with the 3-dimensional case (cf. [5]), where rotationallysymmetric totally geodesic free boundary minimal surfaces have Morse indexone; we prove in this work that the Morse index of a free boundary rotationallysymmetric totally geodesic hypersurface of the n -dimensional RiemannnianSchwarzschild space with respect to variations that are tangential along thehorizon is zero, for n ≥ n ≥
8, with Morse index equalto 0. Also, it is shown that, for n ≥
4, there exist infinitely many non-compactfree boundary minimal hypersurfaces, which are not congruent to each other,with infinite Morse index.We also study the density at infinity of a free boundary minimal hypersur-face with respect to a minimal cone constructed over a minimal hypersurfaceof the unit Euclidean sphere. We obtain a lower bound for the density in termsof the area of the boundary of the hypersurface and the area of the minimalhypersurface in the unit sphere. This lower bound is optimal in the sense thatonly minimal cones achieve it. Introduction
We consider, for each n ≥ m >
0, the n -dimensional domain M n := { x ∈ R n ; | x | ≥ R } , being R := (cid:0) m (cid:1) n − , endowed with the Riemannian metric g Sch = (cid:18) m | x | n − (cid:19) n − δ , where | x | = n P i =1 x i and δ denotes the Euclidian metric. We can check that theboundary of M , the horizon S = { x ∈ R n ; | x | = R } , is a closed totally geodesichypersurface in M . Also, we consider the totally geodesic Euclidean hypersurfaceswhich are the rotations of the coordinate hyperplane Σ through the origin minusthe open ball B := B ( R ) of radius R centered at the origin:Σ = { x ∈ R n ; x n = 0 , | x | ≥ R } . (1.1)This hypersurface is a properly embedded free boundary totally geodesic, inparticular minimal, hypersurface in M , i.e., the boundary ∂ Σ coincides with Σ ∩ The first author, Ezequiel Barbosa, is partially supported by Brazilian CNPq (Grant312598/2018-1). The second author, Jos´e M. Espinar, is partially supported by Spanish MEC-FEDER (Grant MTM2016-80313-P and Grant RyC-2016-19359) and Junta de Andaluc´ıa PAIDI(Grant P18-FR-4049). ∂M , its mean-curvature with respect to g Sch vanishes and the boundary ∂ Σ meets ∂M orthogonally.In the 3-dimensional Riemannnian Schwarzschild space, R. Montezuma [5] com-puted the Morse index of Σ , i.e., the maximum number of directions, tangentialalong ∂M , in which the surface can be deformed in such a way that its area de-creases. It was proved that the Morse index of Σ is one. One of the purposes ofthis paper is to compute the Morse index of Σ in the n -dimensional RiemannnianSchwarzschild space, n ≥
4. We obtain the following result:
Theorem 1.1.
The Morse index of Σ ⊂ ( M n , g Sch ) given by (1.1) , up to a spacerotation, is zero. A question that arises here is the following one: Is the hypersurface Σ the onlyone, among properly embedded free boundary minimal hypersurfaces, with Morseindex equal to zero, up to space rotation, in the Schwarszchild space when n ≥ n ≥ Theorem 1.2.
There are properly embedded free boundary minimal hypersurfaces,not totally geodesic, in the n -dimensional Riemannnian Schwarzschild space withMorse index equal to 0 when n ≥ . We even show a more general result that we describe now. Let C Γ := { λy ; y ∈ Γ n − , λ ∈ (0 , ∞ ) } be a cone in R n , n ≥
4, with vertex at the origin, here Γ n − is an embedded closedorientable minimal hypersurface in S n − . We refer to C Γ as the minimal cone over Γ. Finally, consider Σ Γ = { x ∈ C Γ ; | x | ≥ R } . This hypersurface is a properlyembedded free boundary minimal hypersurface in M n . Note that Σ = Σ Γ whenΓ is an equator in S n − . Hence, we show: Theorem 1.3.
The Morse index of the hypersurface Σ Γ ⊂ ( M n , g Sch ) , ≤ n ≤ ,up to a space rotation, is finite if, and only if, Γ is totally geodesic in S n − (i.e., Σ Γ = Σ ); in particular, the Morse index is zero.Remark . Using the techniques of Theorems 1.2 and 1.3, one is able to obtain anupper bound for the maximal annular domain of stability of Σ Γ as in [5, Theorem1.2]. However, in our case, due to the lack of an explicit solution we are not ableto compute the sharp one.As we can see in Theorem 1.3, all non-totally geodesic cones have infinite Morseindex when 4 ≤ n ≤
7. However, for every dimension n ≥
8, we find non-totallygeodesic cones with finite index (in fact, these cones are stable), and infinitely manyother cones with infinite Morse index. This is a consequence of a link between thestability of a general minimal cone Σ Γ and the first eigenvalue of the Jacobi operatorof Γ ⊂ S n − . Theorem 1.4.
Let Σ Γ a cone in the n -dimensional Riemannnian Schwarzschildspace M n , n ≥ . If λ (Γ) + ( n − n − ≥ , where λ (Γ) is the first eigenvalue of the Jacobi operator of Γ in S n − , then Σ Γ isstable. REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 3
Theorem 1.5.
For every n ≥ , there exist infinitely many non-compact freeboundary minimal hypersurfaces in the n -dimensional Riemannnian Schwarzschildspace, which are not congruent to each other, with infinite Morse index.Remark . It would be interesting to study the existence or not of properlyembedded free boundary minimal hypersurfaces in the Riemannnian Schwarzschildspace, n ≥
4, with Morse index one. A good candidate might be to constructrotationally symmetric catenoidal type minimal hypersurfaces.Finally, we consider the density at infinity of a non-compact free boundary min-imal hypersurface Σ with respect to a minimal cone Σ Γ . We obtain an inequalitywhich relates the area of the boundary ∂ Σ of Σ and the density at infinity Θ Γ (Σ).In fact, we show that the ratio between the area of the boundary of ∂ Σ and thearea of Γ is a lower bound for the density at infinity. This is the content of the nextresult.
Theorem 1.6.
The area of the boundary of Σ satisfies area ( ∂ Σ) ≤ m | Γ | Θ Γ (Σ) . (1.2) Moreover, equality holds if and only if Σ is a minimal cone; in such case | Γ | Θ Γ (Σ) = | R − ∂ Σ | , where R − ∂ Σ ⊂ S n − is nothing but the dilation of ∂ Σ into the ( n − − sphere ofradius one. For the three-dimensional case, there is only one free boundary minimal cone,namely Σ . The above theorem, in the case n = 3, was obtained by R. Montezuma[5] with the rigidity part being exactly Σ . In our case, for dimension n ≥
4, wemight consider different minimal cones and, hence, different densities at infinitywith respect to those cones. Therefore, the rigidity part depends on the minimalhypersurface Γ used in the construction of the cone and the density which is beingconsidered. However, we can use the recent resolution of the Willmore conjecturedue to F. Marques and A. Neves [4] to obtain a more specific rigidity result when n = 4: Theorem 1.7.
Let Σ be a properly embedded free boundary minimal hypersurfacein ( M , g Sch ) . Assume that there exists an embedded minimal hypersurface Γ ⊂ S so that achieves the equality in (1.2) and | Γ | Θ Γ (Σ) ≤ π . Then, up to a rotation,either Σ ≡ Σ or Σ is a minimal cone over a Clifford torus. In any dimension, we can also obtain a more specific rigidity result using Allard’sRegularity Theorem [1].
Theorem 1.8.
Let Σ be a properly embedded free boundary minimal hypersurfacein ( M n , g Sch ) . There exists a constant ǫ ( n ) > , depending on the dimension, sothat if there exists an embedded minimal hypersurface Γ ⊂ S n − that achieves theequality in (1.2) and | Γ | Θ Γ (Σ) < ω n − + ǫ ( n ) , where ω n − is the volume of S n − ;then, up to a rotation, Σ ≡ Σ . Preliminaries
Let Σ be a properly embedded hypersurface in ( M n , g Sch ). Consider the class of C vector fields X on M n that are tangential along ∂M and compactly supported.We use { ϕ ( t, · ) } to denote the one-parameter family of diffeomorphisms associated BARBOSA AND ESPINAR to X , and use it to obtain a variation of Σ with variational vector field X , i.e. weconsider ϕ ( t, Σ) = { ϕ ( t, x ); x ∈ Σ } . The first derivative of the area functional inthe direction of X is given as ddt (cid:12)(cid:12) t =0 area g Sch ( ϕ ( t, Σ)) = Z Σ g Sch ( X, H ) dv Σ + Z ∂ Σ g Sch ( X, ν ) ds, where H and ν denote the mean curvature and outward pointing unit co-normalvectors of ∂ Σ, respectively. We also denote by g the induced metric on Σ. Itfollows from this formula that free boundary minimal hypersurfaces are preciselythe critical points of the area functional with respect to tangential variations.Let ξ denote a globally defined unit normal vector field along Σ, which existssince Σ is properly embedded. From now on, we restrict our attention to smoothvariational vector fields that are normal to Σ, i.e., along the hypersurface X = uξ for some smooth function u : Σ → R . The free boundary condition implies that X is an admissible tangential variational vector field. Assuming that Σ is properlyembedded and minimal ( H = 0), the second derivative of the area functional canbe computed as d dt (cid:12)(cid:12) t =0 area g ( ϕ ( t, Σ)) = Q Σ ( u, u ) , where Q Σ ( · , · ) is the quadratic form given by Q Σ ( u, u ) = − Z Σ u (cid:0) ∆ Σ u + (Ric( ξ, ξ ) + | A Σ | ) u (cid:1) dv Σ + Z ∂ Σ u (cid:18) ∂u∂ν − A ∂M ( ξ, ξ ) u (cid:19) ds , (2.1)where Ric is the Ricci curvature of ( M n , g Sch ) and we are using the notations A Σ and A ∂M for the second fundamental forms of Σ and ∂M , respectively: A ∂M ( V, W ) = − g Sch ( ¯ ∇ η V, W )where η is the inwards pointing unit normal along ∂M , and A Σ ( V, W ) = − g Sch ( ¯ ∇ ξ V, W ) . Fix
R > R , as in [5], we make the following definition: Definition 2.1.
Let Σ( R ) = Σ ∩ {| x | ≤ R } . The Morse index for functionsvanishing on {| x | = R } , Ind F (Σ( R )) , of Σ( R ) is defined as the maximal dimensionof a linear subspace V of smooth functions u : Σ( R ) → R vanishing on {| x | = R } such that Q Σ ( u, u ) < , for all u ∈ V \ { } .The Morse index, Ind(Σ) , of Σ is defined as Ind(Σ) := lim sup R → + ∞ Ind F (Σ( R )) , possibly being infinite. Moreover, when Ind(Σ) = 0 , we say that Σ is stable. Equivalently, the Morse index Ind F (Σ( R )) is the number of negative eigenvalues,counting multiplicities, of the problem (cf. [5, Definition 2.1])( F ) J Σ ψ = − βψ in Σ( R ) ψ = 0 on S ( R ) ∩ Σ ∂ψ∂ν = 0 on S ∩ ∂ Σ REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 5 where J Σ ψ := ∆ Σ ψ + (Ric( ξ, ξ ) + | A Σ | ) ψ and S ( R ) = S n − ( R ) is the spherecentered at the origin and radius R . Henceforth, we will consider n ≥
4. The case n = 3 was considered in [5].2.1. Fischer-Colbrie Criterion.
A first task to do is to characterize the stability,Ind(Σ) = 0, in terms of subsolutions of the differential equation J Σ u ≤
0, this isknown as the Fischer-Colbrie Criterion and its proof follows from the original onegiven in (cf. [3]); however we include it here for the sake of completeness.
Lemma 2.1 (Fischer-Colbrie Criterion) . Let Σ be a properly embedded free bound-ary minimal hypersurface in the n -dimensional Schwarszchild Riemannian manifold ( M n , g Sch ) , n ≥ . Let J Σ the Jacobi operator of Σ .If there is a smooth positive function u on Σ such that ( ∗ ) ( J Σ u ≤ in Σ ,∂u∂ν = 0 on S ∩ ∂ Σ , then Σ is stable, i.e. Ind(Σ) = 0 .Proof.
Assume there exists a smooth positive function u in Σ satisfying ( ∗ ). Given R > R , take the first eigenvalue λ ( R ) and the first eigenfunction f on Σ( R )associated with λ ( R ): J Σ f = − λ ( R ) f in Σ( R ) f = 0 on S ( R ) ∩ Σ ∂f∂ν = 0 on S ∩ ∂ Σ . Set ψ = fu . Note that uψ = 0 on S ( R ) ∩ ∂ Σ and ∂uψ∂ν = 0 on S ∩ ∂ Σ. Hence,integrating by parts, we obtain that λ ( R ) Z Σ( R ) f dv = Z Σ( R ) |∇ Σ f | − f (Ric( ξ, ξ ) + | A Σ | ) dv Σ = Z Σ( R ) |∇ Σ ( uψ ) | − ( uψ ) (Ric( ξ, ξ ) + | A Σ | ) dv = Z Σ( R ) − uψ ∆ Σ ( uψ ) − (Ric( ξ, ξ ) + | A Σ | ) u ψ dv + Z ∂ Σ( R ) uψ ∂uψ∂ν ds = Z Σ( R ) uψ ( − ∆ Σ u − u (Ric( ξ, ξ ) + | A Σ | )) − uψ g ( ∇ Σ u, ∇ Σ ψ ) − u ψ ∆ Σ ψdv ≥ Z Σ( R ) − uψ g ( ∇ Σ u, ∇ Σ ψ ) − u ψ ∆ Σ ψdv = Z Σ( R ) |∇ Σ ψ | u − div Σ ( u ψ ∇ Σ ψ ) dv = Z Σ( R ) |∇ Σ ψ | u dv ≥ . Hence, λ ( R ) ≥
0. This is enough to obtain that
Ind F (Σ( R )) = 0, and, conse-quently, Ind (Σ) = 0. (cid:3)
Remark . We will use the Fischer-Colbrie Criterion in order to prove Theorems1.1 and 1.2.
BARBOSA AND ESPINAR
Cones in the Schwarschild space.
Let us consider now a class of minimalhypersurfaces in the Schwarschild space. Let C Γ := { λy ; y ∈ Γ n − , λ ∈ (0 , ∞ ) } be a cone in R n , n ≥
4, with vertex at the origin. Here Γ n − is an embeddedclosed orientable minimal hypersurface in S n − , the standard ( n − − dimensionalsphere centered at the origin. We refer to C Γ as the minimal cone over Γ. Notethat Σ = Σ Γ , where Γ is an equator in S n − .Observe that the support function ρ = h x, ξ i , x ∈ C Γ , satisfies ρ ≡ C Γ .Hence, Σ Γ = { x ∈ C Γ ; | x | ≥ R } is a properly embedded free boundary minimalhypersurface in ( M n , g Sch ) if, and only if, it is a properly embedded free boundaryminimal hypersurface in ( R n \ B , δ ), where δ the Euclidean metric. Example 2.1 (Clifford cones) . As an example, consider the Clifford torus T m,n − := S m ( λ ) × S ( n − − m ( λ ) , where λ = q mn − , λ = q ( n − − mn − and ≤ m ≤ n − . Since T m,n is a minimalhypersurface in S n − , the cone C m,n = { λy ; y ∈ T m,n − , λ ∈ ( R , ∞ ) } is a mini-mal hypersurface in the Riemannian Schwarzschild space ( M n , g Sch ) . Here, S p ( λ ) denotes the p − dimensional sphere of Euclidean radius λ centered at the origin. Relating the Schwarzschild and Euclidean geometries of cones.
Let Σ := Σ Γ be a cone as defined above. The Schwarzschild metric is conformal to the Euclidanmetric by g Sch = f δ , where f ( | x | ) = (cid:18) m | x | n − (cid:19) n − , (2.2)then, if we denote by g and g δ the induced metric on Σ in the Schwarzschild andEuclidean metric respectively, we can observe that both metrics are conformal andrelated by g := F N − g δ , where F = (cid:18) m | x | n − (cid:19) N − n − and N = n − . (2.3)Also, we can relate the second fundamental forms using [6, Lemma 10.1.1] andthat Σ is a cone, specifically | A (Σ ,g Sch ) | = | A (Σ ,δ ) | F − N − , (2.4)where A (Σ ,g Sch ) and A (Σ ,δ ) are the second fundamental forms of Σ as a hypersurfacein the Schwarschild and Euclidean metric respectively. Recall that, since Σ is min-imal, | A (Σ ,δ ) | = − S (Σ ,δ ) is nothing but the scalar curvature of Σ as a hypersurfacein the Euclidean space ( R n \ B , δ ).Finally, using the Yamabe equation (cf. [8, Section 1]) for the conformal metrics(2.3) and | A (Σ ,g Sch ) | = − S (Σ ,δ ) F − N − , we obtain − F − N +2 N − ∆ δ F = N − N − (cid:0) S Σ + | A (Σ ,g Sch ) | (cid:1) , (2.5)where ∆ δ denotes the Laplacian with respect to the metric g δ REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 7 Bounds on the Morse index
Since round spheres S ( R ) ≡ S n − ( R ), R ≥ R , are totally umbilic in the Eu-clidean space and the Schwarzschild metric is conformal to the Euclidean metric,it follows that S ( R ) is totally umbilic in the Schwarszchild space. In particular(cf. [6, Lemma 10.1.1]), one can easily see that the second fundamental form of S ( R ) with respect to the outer unit normal in the Schwarschild space is given by A S ( R ) ( ν, ν ) = − κ ( R ) g Sch ( ν, ν )for every ν ∈ T x S ( R ), | x | = R and R ≥ R , where κ ( R ) = ( R n − − R n − ) R ( R n − + R n − ) nn − . For each
R > R , consider the compact domain Ω( R ) = B ( R ) \ B . Sincea properly embedded free boundary minimal hypersurface Σ in ( M n , g Sch ) hasboundary ∂ Σ ⊂ S , then Σ( R ), the connected component of Σ ∩ Ω( R ) whoseboundary contains ∂ Σ, is a properly embedded minimal hypersurface in Ω( R ) whoseboundary satisfies ∂ Σ( R ) ⊂ S ∪ S ( R ). Note also that along the components ofthe boundary ∂ Σ( R ) in S , Σ( R ) meets S orthogonally. However, the componentsof the boundary ∂ Σ( R ) in S ( R ) might fail to satisfy this orthogonality condition.This means that it can happen that Σ( R ) is not a properly embedded free boundaryminimal hypersurface in Ω( R ).When Σ( R ) is a properly embedded free boundary minimal hypersurface in Ω( R ),i.e., along all the components of the boundary ∂ Σ( R ), Σ( R ) meets the boundary ∂ Ω( R ) orthogonally (this is the case when Σ is a cone), we can consider the Morseindex (quadratic) form, Q Σ ( R ), of Σ( R ) given by Q Σ ( R )( ψ, ϕ ) = − Z Σ( R ) ψ (cid:0) ∆ ϕ + (Ric( ξ, ξ ) + | A Σ | ) ϕ (cid:1) dv Σ + Z ∂ Σ( R ) ψ (cid:18) ∂ϕ∂ν − qϕ (cid:19) ds , (3.1)where q = (cid:26) S ,κ ( R ) on S ( R ) . It is worth to mention here that, for every
R > R , we always have the quadraticform Q Σ related with the index Ind F (Σ( R )) (see Definition 2.1). If Σ( R ) is alsoa free boundary minimal hypersurface in Ω( R ), we also have the quadratic form Q Σ ( R ) (given by (3.1)). In this case, let us denote by Ind M (Σ( R )) the Morse indexof Σ( R ) as a free boundary minimal hypersurface with respect to the quadraticform Q Σ ( R ). Observe that: Claim A:
Ind F (Σ( R )) ≤ Ind M (Σ( R )) .Proof of Claim A. In fact, if ψ ∈ V , where V is the space spanned by the eigen-function ψ i of ( F ), with β i <
0, we obtain Q Σ ( R )( ψ, ψ ) = Q Σ ( ψ, ψ ) <
0; thisproves Claim A. (cid:3)
Moreover, since q ≥ R ), as a free boundary minimal hypersurface in Ω( R ),is given by the addition Ind M (Σ( R )) = Ind D (Σ( R )) + Null D (Σ( R )) + Ind R (Σ( R )), BARBOSA AND ESPINAR where Ind D (Σ( R )) is the number of non-positive eigenvalues, counting multiplicity,of the problem( D ) (cid:26) ∆ Σ v + (Ric( ξ, ξ ) + | A Σ | ) v = − δv in Σ( R ) ,v = 0 on ∂ Σ( R ) , Null D (Σ( R )) is the nullity of the above problem, and Ind R (Σ( R )) is the number ofeigenvalues smaller than 1, counting multiplicity, of the problem( R ) ( ∆ Σ u + (Ric( ξ, ξ ) + | A Σ | ) u = 0 in Σ( R ) ,∂u∂ν = λqu on ∂ Σ( R ) . Reasoning as above, we can also easily show:
Claim B:
Ind D (Σ( R )) ≤ Ind F (Σ( R )) . Relating the Euclidean and Schwarschild index forms.
From now on,Σ will always denote a minimal cone Σ := C Γ ∩ {| x | ≥ R } . As we have pointedout above, a minimal cone, Σ, in the Schwarshild metric is also minimal in theEuclidean metric, and viceversa. Hence, we will relate the Schwarschild index form Q Σ ( R ) and the Euclidean index form Q δ ( R ) for the Dirichlet problem; i.e., Q δ ( R )( ψ, ϕ ) = − Z Σ ∩ Ω( R ) ψ (cid:0) ∆ δ ϕ + | A Σ | ) ϕ (cid:1) dδ, ψ = 0 on ∂ Σ( R ); (3.2)where dδ is the area element associated to g δ . Lemma 3.1. If ψ is zero on ∂ Σ( R ) , R > R , then Q Σ ( R )( F − ψ, F − ψ ) = Q δ ( R )( ψ, ψ ) − NN − Z Σ( R ) (cid:0) F − ∆ δ F (cid:1) ψ dδ. (3.3) Proof.
First, it follows from the Gauss equation thatRic( ξ, ξ ) + | A (Σ ,g ) | = − S Σ | A (Σ ,g ) | Σ is the scalar curvature of (Σ , g ). Second, denote by L g the conformalYamabe operator of the metric g on Σ given by L g u = ∆ Σ u − N − N −
1) S Σ u , (3.5)Hence, for any smooth function u ∈ C ∞ (Σ), we obtain∆ Σ u + (Ric( ξ, ξ ) + | A (Σ ,g ) | ) u = (3.5) L g ( u ) + N − N −
1) S Σ u + (Ric( ξ, ξ ) + | A (Σ ,g ) | ) u = (3.4) L g ( u ) − N N −
1) S Σ u + | A (Σ ,g ) | u = (2.5) L g u + NN − (cid:16) F − N +2 N − ∆ δ F (cid:17) u + (cid:18) − N − N − (cid:19) | A (Σ ,δ ) | F − N − u, where we have used (2.4). On the other hand, since g = F N − g δ , it follows (cf. [8,Section 1]) that L g ( F − ψ ) = F − N +2 N − L δ ( ψ ) = F − N +2 N − (cid:18) ∆ δ ψ − N − N −
1) S (Σ ,δ ) ψ (cid:19) . REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 9
Therefore, using the above two equations and (2.4), the Jacobi operator J Σ v =∆ Σ v + (Ric( ξ, ξ ) + | A (Σ ,g ) | ) v on Σ satisfies J Σ ( F − v ) = L g ( F − v ) + NN − (cid:16) F − NN − ∆ δ F (cid:17) v + (cid:18) − N − N − (cid:19) | A (Σ ,δ ) | F − N +2 N − v = F − N +2 N − (cid:18) ∆ δ v − N − N −
1) S (Σ ,δ ) v (cid:19) + NN − (cid:16) F − NN − ∆ δ F (cid:17) v + (cid:18) − N − N − (cid:19) | A (Σ ,δ ) | F − N +2 N − v = F − N +2 N − (cid:18)(cid:0) ∆ δ v + | A (Σ ,δ ) | v (cid:1) + NN − (cid:0) F − ∆ δ F (cid:1) v (cid:19) . Consider the operator J δ v = (cid:0) ∆ δ + | A (Σ ,δ ) | (cid:1) v + NN − (cid:0) F − ∆ δ F (cid:1) v, (3.6)hence we can re-write the above Jacobi operator as v J δ ( v ) = ( F − v ) J Σ ( F − v ) F NN − . Thus, since the volume elements of g and g δ are related by dv Σ = F NN − dδ from(2.3), we have obtained that the index forms Q Σ ( R ) and Q δ ( R ), given by (3.1) and(3.2) respectively, satisfy (3.3) for every ψ that vanishes on ∂ Σ( R ) as claimed. (cid:3) Finally, we must control the last term in the above equation (3.3). Hence, NN − F − ∆ δ F = NN − F − ( d Fdr + N − r dFdr ) = ( n − m r n − ( m + 2 r n − ) , for all x ∈ Σ such that | x | = r ≥ R . Thus, Lemma 3.1 and the above observationimplies Lemma 3.2. If ψ is zero on ∂ Σ( R ) , R > R , then Q Σ ( R )( F − ψ, F − ψ ) = Q δ ( R )( ψ, ψ ) − ( n − m Z Σ( R ) r n − ( m + 2 r n − ) ψ dδ . Euclidean index form over cones.
In this part we follow the seminal workof J. Simons [9]. On the one hand, fix R < R and consider the following initialvalue problem on the interval [ R , R ]:(IVP) − r d gdr − ( n − r dgdr = β g in ( R , R ) ,g ( R ) = 0 = g ( R ) . From [9, Lemma 6.1.5], for each j ∈ N , the function g j ( r ) = c j r − ( n − sin (cid:18) jπ log( R/R ) log( r/R ) (cid:19) , where c − j = log( R/R )2 j , solves (IVP); that is, − r d g j dr − ( n − r dg j dr = β j g j , where β j = (cid:18) n − (cid:19) + (cid:18) jπ log( R/R ) (cid:19) . Moreover, let C ∞ ([ R , R ]) denote the space of smooth functions on [ R , R ] whichvanish at the end points. Then, we can obtain a basis on this space by eigenfunc-tions { g j } of (IVP), with eigenvalue β j , which are orthonormal with respect to the L ([ R , R ] , t n − dt )-norm.On the other hand (cf. [9, Lemma 6.1.4]), given Γ ⊂ S n − a minimal hypersurface,consider the Jacobi operator(J) J Γ f := − ∆ Γ f − | A Γ | f ;and denote by f i , i ∈ N , the eigenfunctions of the above operator with eigenvalue λ i . Then, we can obtain a base on the space of smooth functions on Γ, C ∞ (Γ), byeigenfunctions { f i } , with eigenvalue λ i , which are orthonormal with respect to the L (Γ)-norm.Thus, given ψ ( p, t ) ∈ C ∞ (Σ( R )), where C ∞ (Σ( R )) is the space of smooth func-tions which vanish on ∂ Σ( R ), we obtain that ψ has an unique expansion (cf. [9,Lemma 6.1.6]) as ψ ( p, t ) = X i,j =1 a ij f i ( p ) g j ( t ) , and, using [9, Lemma 6.1.3], we obtain Q δ ( R )( ψ, ψ ) = X i,j =1 a ij ( λ i + β j ) . (3.7)3.3. Spectrum of J δ in spherical coordinates. Following [9], using separationof variables ψ ( p, r ) = f ( p ) u ( r ), the expression of the Laplace operator ∆ δ on Σ inspherical coordinates given by∆ δ = ∂ ∂r + N − r ∂∂r + 1 r ∆ Γ , and r | A (Σ ,δ ) | = | A Γ | ; a function ψ ( p, t ) ∈ C ∞ (Σ( R )) solution to J δ ( ψ ) = − λψF N − , where J δ is given by (3.6), must satisfy that • f ∈ C ∞ (Γ) belongs to the spectrum of J Γ ; with eigenvalues λ k (Γ), k ∈ N . • u ∈ C ∞ ([ R , R ]) is a solution to L k ( u ) = − λF N − u , where L k is a family(indexed by k ∈ N ) of Sturm-Liouville operators defined by L k := d dr + N − r ddr + V ( r ) − λ k (Γ) r , where V ( r ) = NN − (cid:0) F − ∆ δ F (cid:1) = m ( n − r n (cid:18) r n − m + 2 r n − (cid:19) . Hence, L k := d dr + N − r ddr + m ( n − r n (cid:18) r n − m + 2 r n − (cid:19) − λ k (Γ) r . (3.8)If we consider the change v ( r ) = r N − u ( r ), we obtain L k u = r − N − (cid:18) d v k dr + W k v k (cid:19) , REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 11 where W k = m ( n − r n (cid:18) r n − m + 2 r n − (cid:19) − λ k (Γ) + ( n − n − r . (3.9) Lemma 3.3.
The positive function v ( r ) := (cid:18) r n − m + 2 r n − (cid:19) n − , r ≥ R , (3.10) satisfies Lv := d vdr + m ( n − r n (cid:18) r n − m + 2 r n − (cid:19) v = 0 . (3.11) Proof.
A straightforward computation shows that v ′ ( r ) = m r n − (cid:18) r n − m + 2 r n − (cid:19) n − +1 and v ′′ ( r ) = − ( n − m r n (cid:18) r n − m + 2 r n − (cid:19) n − +2 , which shows the lemma. (cid:3) Proof of Theorem 1.1
Assume that Γ is totally geodesic, hence λ (Γ) = 0. For n ≥
4, (3.9) implies W ≤ m ( n − r n (cid:18) r n − m + 2 r n − (cid:19) , hence Lemma 3.3, (3.8) and (3.11) imply that u ( r ) = r − N − v ( r ) satisfies L u = r − N − (cid:18) d vdr + W v (cid:19) ≤ r − N − Lv = 0 , where v is given by (3.10). Hence, the function ψ ( p, r ) = F − ( r ) u ( r ) = 2 r n − m + 2 r n − is a positive function on Σ such that J Σ ( ψ ) ≤
0. Also, a straightforward computa-tion shows ∂ψ∂r ( p, r ) = ( n − r n − ( m + 2 r n − ) (cid:0) m − r n − (cid:1) . Since 2 R n − = m , we can check that ∂ψ∂r ( p, R ) = 0. Therefore, the Fischer-Colbrie Criterion, Lemma 2.1, implies that Σ is stable. This proves Theorem 1.1.4.1. Proof of Theorem 1.2.
In the case that Γ is the Clifford torus, we knowthat λ (Γ) = − ( n − n ≥
8, Lemma 3.3, (3.8), (3.9) and (3.11)imply that L u ≤ u ( r ) = r − N − v ( r ) where v is given by (3.10). Hence, usingthe Fischer-Colbrie Criterion we can show that Σ is stable as above. This provesTheorem 1.2.4.2. Proof of Theorem 1.4.
In this case, the condition 4 λ (Γ)+( n − n − ≥ L u ≤ u ( r ) = r − N − v ( r ) where v is given by (3.10). Thus, followingthe above ideas we can prove Theorem 1.4. Proof of Theorem 1.3
Now, let Γ ⊂ S n − be a compact minimal hypersurface in the ( n − − dimensionalsphere. Fix R > R and consider ψ j ( p, r ) := f ( p ) g j ( r ), x = ( p, r ) ∈ Γ × [ R , + ∞ ) = C Γ , which vanishes on ∂ Σ( R ). It is well-known that the first eigenvalue of (J)satisfies λ ≤ − ( n −
2) and β j = (cid:18) n − (cid:19) + (cid:18) jπ log( R/R ) (cid:19) , for every j ∈ N . Hence, Lemma 3.2 and (3.7) imply Q gR ( F − ψ j , F − ψ j ) = ( λ + β j ) − ( n − m Z Σ( R ) r n − ( m + 2 r n − ) ψ j dδ. Using Fubbini and the expression of dδ = r n − drd Γ as a product metric, d Γ thevolume element of Γ ⊂ S n − , we get Q gR ( F − ψ j , F − ψ j ) = ( λ + β j ) + e G j ( R ) , where e G j ( R ) = − ( n − mc j Z RR r n − ( m + 2 r n − ) sin (cid:18) jπ log( r/R )log( R/R ) (cid:19) drr . Consider the change of variable s = π log( R/R ) log( r/R ) and 2 a ( R )( n − π ds = drr ;where a ( R ) = ( n −
2) log(
R/R )2 π , (5.1)then e G j ( R ) := ( n − j π Z π cosh − ( a ( R ) s ) sin ( js ) ds. Hence, following the exact same computations as above we achieve Q gR ( F − ψ j , F − ψ j ) = λ + (cid:18) n − (cid:19) + G j ( R ) , where G j ( R ) := (cid:18) ( n − j a ( R ) (cid:19) − ( n − j π Z π cosh − ( a ( R ) js ) sin ( s ) ds. Therefore, if 4 ≤ n ≤ λ + (cid:0) n − (cid:1) < G j ( R ) → R → + ∞ for all j ≥
1, we obtain that Q Σ ( R )( F − ψ j , F − ψ j ) < j ≥ R large enough depending on j, that is, Ind D (Σ( R )) → + ∞ as R → + ∞ , since the ψ ′ j s are linearly independent.This and Claim B prove Theorem 1.3.5.1. Proof of Theorem 1.5.
This result follows from Theorem 1.3.
REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 13 Density over minimal cones
In this section, following ideas of [2, 5], we represent the Schwarzschild manifoldas M n = S n − × ( s , + ∞ ), n ≥
3, endowed with the metric g Sch = 11 − ms − n ds + s g S n − , where s = (2 m ) n − . We define a continuous function F : [ s , + ∞ ) → R by F ′ ( s ) = √ − ms − n and F ( s ) = 0. Making the change r = F ( s ), the metric g Sch can be rewritten as g Sch = dr + h ( r ) g S n − , where h : [0 , + ∞ ) → [ s , + ∞ ) denotes the inverse of F . Hence, from [2, Section 5],we obtain h ′ ( r ) = p − ms − n , where s = h ( r ). The variable r = r ( x ) represents the Schwarzschild distance to thehorizon ∂M . Moreover, consider the function f ( x ) = h ′ ( r ( x )) and the conformalvector field X = h ( r ) ∂ r , where ∂ r denote the unit length radial vector, i.e., the gradient, with respect to theSchwarszchild metric g Sch , of the function r . Using this explicit expression of X ,we can obtain the divergence of X , on Σ, with respect to the metric g Sch : div Σ ( X ) = ( n − f . In fact, following [2, Section 2], we can write X = ∇ Σ ϕ , for some function ϕ such that Hess Σ ϕ = f g . Hence, div Σ X = n − X k =1 Hess Σ ϕ ( E k , E k ) − Hg ( X, N ) = ( n − f , where { E , ..., E n − } is an orthonormal basis of the tangent space to Σ and H = 0.The function f is nothing but the static potential associated to the Schwarschildmanifold.Let Σ be a properly embedded free boundary minimal hypersurface in ( M n , g Sch ).Let B ρ denote the set of points in ( M n , g Sch ) at Schwarzschild distance to the hori-zon at most ρ . Definition 6.1.
Let Γ be a closed minimal hypersurface in the Euclidean unitsphere S n − . We define the Γ -density at infinity of a properly embedded free bound-ary minimal hypersurface Σ in the Schwarzschild manifold by Θ Γ (Σ) := lim ρ → + ∞ vol (Σ ∩ B ρ ) vol (Σ Γ ∩ B ρ ) whenever this limit exists, where Σ Γ denotes the cone over Γ . Observe that when n = 3 the only minimal cone is the one given by a greatcircle in S . In our case, in higher dimension, we have a plethora of minimal conesin order to consider a density. Denote by | S | the volume of a hypersurface S in theEuclidean unit sphere S n − . With this notation, we can announce the followingresult obtained by R. Montezuma [5] when n = 3. Theorem 6.1. If Θ Γ (Σ) exists and is finite; the following formula is valid: Θ Γ (Σ) = area ( ∂ Σ)2 m | Γ | + n − | Γ | Z Σ fh n − ( r ) | ∂ ⊥ r | g dv . (6.1) Proof.
We follow [5, Section 3] with minor changes due to the dimension. For every0 < σ < ρ , consider the vector field W defined by the expression W ( x ) := (cid:16) h n − ( σ ) − h n − ( ρ ) (cid:17) X ( x ) if x ∈ M with 0 ≤ r ( x ) ≤ σ, (cid:16) h n − ( r ) − h n − ( ρ ) (cid:17) X ( x ) if x ∈ M with σ ≤ r ( x ) ≤ ρ, div Σ W ( x ) = (cid:16) h n − ( σ ) − h n − ( ρ ) (cid:17) ( n − f for 0 ≤ r ( x ) < σ, ( n − fh n − ( ρ ) + ( n − fh n − ( r ) | ∂ ⊥ r | g for σ < r ( x ) < ρ, where ∂ Tr and ∂ ⊥ r denote the tangential and normal components of ∂ r , respectively,relative to the tangent spaces of Σ. Note that div Σ W ( x ) = 0 if r ( x ) > ρ .Therefore, for almost all 0 < σ < ρ , we obtain1 h n − ( ρ ) Z Σ ρ f dv = 1 h n − ( σ ) Z Σ σ f dv + Z Σ ρ \ Σ σ fh n − ( r ) | ∂ ⊥ r | g dv − n − Z ∂ Σ g ( W T , ν ) ds , where W T denotes the tangential component of W . Here, we have used that the di-vergence over Σ of the normal component of W vanishes, since Σ is a free boundaryminimal hypersurface.Note that, since Σ is free boundary, it follows that W is tangential to Σ and ν = − ∂ r . Then, g ( W T , ν ) = − (cid:18) h n − ( σ ) − h n − ( ρ ) (cid:19) h (0) = − (2 m ) n − (cid:18) h n − ( σ ) − h n − ( ρ ) (cid:19) , and, consequently, for 0 ≤ σ < ρ , we have µ (Σ ∩ B ρ ) h n − ( ρ ) = µ (Σ ∩ B σ ) h n − ( σ ) + Z Σ ρ \ Σ σ fh n − ( r ) | ∂ ⊥ r | g dv + (2 m ) n − n − (cid:18) h n − ( σ ) − h n − ( ρ ) (cid:19) area ( ∂ Σ) , (6.2)where area ( ∂ Σ) represents the area of the boundary ∂ Σ, and µ is the measuredefined by µ ( A ) = R A f . Now, if Θ Γ (Σ) exists, we can check thatlim ρ → + ∞ area (Σ ∩ B ρ ) area ( C Γ ∩ B ρ ) = lim ρ → + ∞ ( n − area (Σ ∩ B ρ ) | Γ | h n − ( ρ ) . Therefore, assuming that Θ Γ (Σ) exists, we let σ = 0 and ρ → + ∞ in the identity(6.2) to conclude that (6.1) holds. (cid:3) Since the integral term at the right hand side of (6.1) is non-negative, we obtain:
REE BOUNDARY MINIMAL HYPERSURFACES IN THE SCHWARZSCHILD SPACE 15
Corollary 6.1.
The area of the boundary of Σ satisfies area ( ∂ Σ) ≤ m | Γ | Θ Γ (Σ) . (6.3) Moreover, equality holds if and only if Σ is a minimal cone; in such case | Γ | Θ Γ (Σ) = | R − ∂ Σ | , where R − ∂ Σ ⊂ S n − is nothing but the dilation of ∂ Σ into the ( n − − sphere ofradius one.Proof. The inequality follows from (6.1). In the case of equality, Σ must be a coneand, in particular, ∂ Σ is a minimal hypersurface in ( S ( R ) , g Sch ). We can easilycompute area ( ∂ Σ) = 2 m | R − ∂ Σ | , which finishes the proof. (cid:3) At this point, we might consider different minimal cones in order to establishrigidity results in terms of the density6.1.
Proof of Theorem 1.7.
Since we are assuming equality in (6.3), Σ mustbe a cone and, in particular, ∂ Σ is a minimal hypersurface in ( S ( R ) , g Sch ) and | R − ∂ Σ | ≤ π . Hence, [4] implies that R − ∂ Σ ⊂ S is either a great sphere or aClifford torus, which proves this theorem.6.2. Proof of Theorem 1.8.
In any dimension, by the Monotonicity Formula andAllard’s Regularity Theorem [1], we can obtain Theorem 1.8.
References [1] W. K. Allard,
On the first variation of a varifold , Ann. of Math., (1972), 417–491.[2] S. Brendle, Constant Mean Curvature surfaces in Warped Product Manifolds , Publ. Math.de l’IH´ES, (2013), 247–269.[3] D. Fischer-Colbrie,
On complete minimal surfaces with finite Morse index in three mani-folds , Invent. Math., (1985), 121–132.[4] F. Marques, A. Neves, Min-max theory and the Willmore conjecture , Annals of Mathe-matics, (2014), 683–782.[5] R. Montezuma,
On free boundary minimal surfaces in the Riemannian Schwarzschildmanifold . To appear in
Bull. Braz. Math. Soc. [6] R. L´opez,
Constant Mean Curvature Surfaces with Boundary . Springer Monographs inMathematics. 2013.[7] H. Tran, D. Zhou,
On the Morse Index with Constraints I: An Abstract Formulation .Arxiv.[8] J. Escobar,
Uniqueness Theorems on Conformal Deformation of Metrics, Sobolev Inequal-ities, and an Eigenvalue Estimate , Communications on Pure and Applied Mathematics, (1990), 857–883.[9] J. Simons, Minimal Varieties in Riemannian Manifolds , The Annals of Mathematics,Second Series, (1968) no. 1 , 62–105. Departamento de Matem´atica, Universidade Federal de Minas Gerais, Belo Horizonte-Brazil
Email address : [email protected] Universidad de C´adiz - Spain
Email address ::