Dirac operator and hypersurfaces
Sebasti´an Montiel
Departamento de Geometr´ia y Topolog´ia, Facultad de Ciencias Universidad de Granada, 18071 Granada, Spain
e-mail : [email protected]
(2000Mathematics Subject Classification: 53A10, 53C42.)
1 Introduction
First of all, I would like to thank the organizer of the 9th Workshop on Differen- tial Geometry, held in Taegu, Professor Suh, for his kind invitation to participate.
Also, I thank Professor Suh and his students for their great hospitality, during my stay at Kyungpook National University.
In this paper I want to expose some ideas and results appearing on the inter- action field between two topics of the Global Differential Geometry: Geometry of Submanifolds and Spectral Theory. The results I am going to speak about are part of different joint works with O. Hijazi, from the University of Nancy (France), with X. Zhang, from the Academy of Sciences in Beijing (China) and with my student in Granada A. Roldn.
2 Spectral results about the Laplacian
Research works about spectral properties of submanifolds in Riemannian man- ifolds have been frequent since the seventies [BlW, Ch, CW, MR, O, Re1, Re2, Ta].
These spectral properties are usually referred to the Laplacian operator associated to the metric on the submanifold induced from the Riemannian structure of the ambient space and they mainly deal with two classes of problems:
Firstly, to obtain information about the spectrum of the Laplacian on a compact submanifold M immersed in a given Riemannian ambient space, which is almost always a Riemannian manifold with a well-behaved curvature: Euclidean space, sphere, projective space and other symmetric spaces. One looks for information given in terms of the extrinsic geometry of the submanifold, that is, in terms of its second fundamental form, its mean curvature, etc. From a physical point of view, we are trying to hear the shape of the submanifold [GWW, K], that is, we would like to know which is the relation between the pure sounds given off by the submanifold and the way in which it is immersed into the ambient space.
Secondly, to characterise some distinguished submanifolds (with an easier ge- ometry or second fundamental form) by a limiting behaviour of their eigenvalues,
1
their multiplicities or their eigenfunctions.
Let me to be precise by setting two examples of this kind of results. A theorem due to Bleecker and Weiner in 1976 [BlW] (and under different forms to Reilly and Chen [Re2, Ch]) says that the first non-null eigenvalue λ1 of the Laplacian of a compact hypersurface M immersed into the (n+ 1)-dimensional Euclidean space Rn+1 satisfies the following inequality
λ1≤n Z
M
H2dV vol(M) ,
where vol(M) is the volume of the hypersurface andH denotes its mean curvature function. And we have the equality if and only if the hypersurface is a (round) hypersphere.
Another meaning example could be the following one. It is an easy and old observation by Takahashi [Ta] that the position vector functionφ:M →Sn⊂Rn+1 of a minimal hypersurface immersed into the unit sphere is an eigenfunction of the Laplacian of its induced metric corresponding to the eigenvalue n = dimM. Yau and Ogiue asked in 1982 [Y, O] the following question: what can you say about compact minimal hypersurfaces immersed into a unit sphere for which their dimension n is just the first non-null eigenvalue? This type of hypersurfaces are usually called minimal hypersurfaces immersed by the first eigenfunctions. Yau even conjectured [Y] that this must be always the case when the hypersurface is embedded. At the moment we do not know if this conjecture is true. The best approach is by Choi and Wang [CW] who showed in 1983 that the first non-null eigenvalue of the Laplacian on an embedded minimal hypersurface in the (n+ 1)- dimensional unit sphere is bigger thann/2.
A. Ros and myself studied in 1986 [MR] the case of the three-sphere. It was a well-known result by Almgren [L] that a minimal surface in S3 with genus zero must be an equator. We proved that if the genus is one and the surface is immersed by the first eigenfunctions, then it is congruent to the so called Clifford torus
{(x1, x2, x3, x4)∈S3|x21+x22=x23+x24= 1/2}.
In this way we saw that Yau’s conjecture was equivalent, for tori, to another well- known conjecture put down to Hsiang and Lawson [HL] which claims that this Clifford torus is the only minimal torus in the three-sphere without self-intersections.
We do not know anything yet for higher genera.
I think that these problems and results can help us to understand what kind of things attract the interest in the spectral theory of submanifolds, at least when the involved operator is the Laplacian.
3 The classical Dirac operator
In the last years, some geometers of submanifolds (like myself or the group of the Technische Universitt of Berlin: Ferus, Pinkall, Taimanov and others
[FLPP, HM, HMR2, HMZ1, HMZ3, T]) have been led to study diverse Dirac operators which arise from the extrinsic geometry of surfaces, and some Rie- mannian geometers specialists in this operator (like B¨ar, Friedrich or Hijazi [AFr, Am, B¨a, Bur, HM, HMR2, HMZ1, HMZ3]) have started to consider its par- ticular features on submanifolds.
The Dirac operator is an elliptic differential operator of order one playing an increasing central role both in modern physics and mathematics. However, it did not appear in the context of Riemannian geometry until 1962. In that year, Atiyah and Singer [AS] were able to define a non-local version of this operator on some orientable Riemannian manifolds. Notice that still in 1937 lie Cartan [C] in his bookLa th´eorie des spineurstalked about the unsurmountable difficulties to apply techniques of global calculus to spinors and Dirac operators on Riemannian and Lorentzian manifolds with non-trivial curvature.
The classical Dirac operator had been invented in 1928 by the physicist P.A.M.
Dirac [D] as a square root of the usual wave or D’Alembert operator. His aim was to give a quantum-mechanical description of electrons taking into account the recent relativistic laws. He had to find a way to do compatible the relativistic Klein- Gordon or wave equation and the Schr¨odinger equation of the classical quantum mechanics. The first one (I have written the mass-free versions)
¤u=−∂2u
∂t2 + ∆u= 0 u:R×M →R M ⊂R3
is invariant under Lorentz transformations and is of second order in all the variables and its solutions can be expanded, under suitable boundary conditions, into a series ofpure sounds
u(t, p) =f(t)φ(p) f00+λf = 0 ∆φ+λφ= 0 λ≥0 involving eigenvalues and eigenfunctions of the Laplacian. The second one
−i∂ψ
∂t + ∆ψ= 0 ψ:R×M →C
is invariant under Galileo transformations and is of first order in the time and its solutions are expanded into series ofpure states
ψ(t, p) =f(t)φ(p) f0+iλf = 0 ∆φ+λφ= 0 λ≥0.
Notice that the solutions take now complex values so that the corresponding expo- nential functions may be bounded. As Einstein’s Special Relativity gave a certain equivalence between space and time coordinates, Dirac looked for a first order equa- tion in the four variables
−i∂ψ
∂t +Dψ= 0 Dψ= X3 i=1
Ai
∂ψ
∂xi
whose iteration should give the wave operator. But it is equivalent to D2 =−∆
and this holds when the coefficientsAi are chosen to verify AiAj+AjAi=−2δij i, j= 1,2,3
It is clear that there are not any complex numbers satisfying these relations. But there are square complex matrices of order two. For instance, the Pauli matrices
A1=
µ i 0 0 −i
¶
A2=
µ 0 1
−1 0
¶
A3=
µ 0 i i 0
¶ .
Then the solutions to the so-called Dirac equation Dψ = 0 (more precisely Weyl equation, in this version) are not either real nor complex functions. They must be maps taking their values in the complex vector spaceC2and are calledspinor fields because they had been already used by Pauli [P] in 1927 to propose a model for thespin, that is, the inner angular momentum of electrons. The physicists found however difficulties in combining Dirac’s first order equation for the relativistic electron with the needs of General Relativity, since the spinors did not transform like vectors or tensors when one changes coordinates in the spaceR3 and so, in a first approach, they had not a geometrical meaning. This was the problem solved by Atiyah and Singer [AS] in 1962 by using as a tool the machinery of principal bundles and their connections which had been invented in the fifties. Using this language, the setting above about spinors and Dirac’s equation could be translated as follows (see, for example, [BFGK, BHMM, Fr2, LM]). We have a complex vector bundle SM = M ×C2 on the open set M ⊂ R3 (in fact a trivial one) endowed with a Hermitian product (the usual one on the fiber) and a covariant derivative∇ (also the trivial one) which parallelizes that metric. Then the spinors solutions to Dirac’s equation can be thought of as sections of this trivial bundle. Also Pauli’s matrices Ai provide a point-wise bundle multiplication of vectors by spinors γ : T M =M×R3→End(SM) =M×gl(2,C) given by
γ(u1, u2, u3) =u1A1+u2A2+u3A3=
µ iu1 u2+iu3
−u2+iu3 −iu1
¶
acting by means of skew-Hermitian traceless endomorphisms. It is called aClifford multiplicationbecause of the relations
γ(u)γ(v) +γ(v)γ(u) =−2hu, viI2
which imply that it extends to a representation of the Clifford algebra constructed onR3on the vector space C2. This representation is irreducible and is compatible with the covariant derivative in the sense that
∇uγ(X)ψ=γ(∇uX)ψ+γ(X)∇uψ,
where the second ∇ stands for the (trivial) connection on vector fields defined on the Euclidean space. In this frame the Dirac operator D is the contraction of ∇
andγ, that is
Dψ= X3 i=1
γ(ei)∇eiψ
where e1, e2, e3 is any orthonormal basis of the Euclidean space.
4 Spinor manifolds and Dirac operator
As I said, Atiyah and Singer found a weak topological obstruction so that an orientable manifold M could support a geometrical structure such as I have just described on the domains of Euclidean three-space. That is, in contrast to the case of the Laplacian, we cannot define the Dirac operator on each Riemannian manifold, but only on the so calledspin manifolds, those whose second Stiefel-Whitney class vanishes. Anyway, it is important to point out from now that the most common ambient spaces in Geometry of Submanifolds, such that Euclidean spaces, spheres, odd-dimensional real or complex projective spaces carry this type of structure.
On such manifolds we have a Hermitianspinor bundleSM with a metric con- nection∇and a Clifford multiplicationγ:T M →EndC(SM), arising an irreducible representation of the corresponding Clifford algebra, with the same compatibility equations you have seen on the screen for the Euclidean case. With these ingre- dients, we may define the Dirac operator on the Riemannian spin manifold M as mapping smooth sections of the spinor bundle in the following way
Dψ= Xn
i=1
γ(ei)∇eiψ
where the sum runs over an orthonormal frame tangent to M. It is clearly a differential operator of first order and two properties follow immediately from the definition. It is an elliptic operator and it is formally self-adjoint with respect to theL2-product of spinor fields. This means that, if the manifoldM is compact, the Dirac operator has a discrete real spectrum
−∞ . · · ·< λ−k<· · ·< λ−1< λ0= 0< λ1<· · ·< λk <· · · %+∞
where λ0 = 0 does not necessarily belong to the spectrum. The main goal of the spectral theory of the Dirac operator is to get information about this sequence in terms of the geometry of M. This topic was initiated by Lichnerowicz in the sixties and continued by Hitchin, Bourguignon, Friedrich, Hijazi, B¨ar and others [Fr1, Hi1, Hi2, Li].
5 Dirac operator on hypersurfaces
Our interest now is just about the relation between the spectral properties of the classical Dirac operator and the geometry of hypersurfaces, in the sense of
the previous results about the Laplacian. So, we will suppose from now on that the manifold M that we are going to deal with is a hypersurface immersed in an (n+ 1)-dimensional Riemannian manifoldM.
If this ambient manifold is spin (and this is the case for the most common ambient spaces in Theory of Submanifolds) and the hypersurfaceM is orientable, thenM is another spin manifold and so we will have also onM a spinor bundleSM, a Levi-Civita spinorial connection, a Clifford multiplication and an (intrinsic) Dirac operatorDM. The point is to relate this spinorial structure on the hypersurface with the restriction to the hypersurface itself of the spinorial structure of the ambient manifold. Without entering into details (one can consult, for instance, [B¨a, BFGK, BHMM, HMZ1, HMZ3]), we can check the following. LetSM be the spinor bundle of M, γ its Clifford multiplication, ∇ its Levi-Civita connection and D its Dirac operator and consider the restriction
SM|M
on the hypersurfaceM of the spinor bundle of the ambient space. If the dimension ofM is even, then this Hermitian bundle can be identified with the intrinsic spinor bundle SM ofM. Instead, if the dimension of M is odd, we may identify it with two copies of thisSM. As soon as we have made these identifications, you can prove that the first order operatorD/ defined for sections of the restricted bundle by
D/ ψ= n
2Hψ−γ(N) Xn i=1
γ(ei)∇eiψ,
where {e1, . . . , en} is an orthonormal frame tangent to the hypersurface, coincides with the Dirac operatorDM of the hypersurface, if its dimension is even, and with the pair (DM,−DM) if the dimension is odd. Then the spectrum of this D/ is nothing but the symmetrization with respect to zero of the spectrum ofDM.
6 Upper estimates on hypersurfaces
Looking at this equality and accepting that the first order operator D/ can be thought of as the intrinsic Dirac operator on the hypersurface M, you can see that the mean curvature function H must play an important role to control its eigenvalues. In fact, a first consequence of it is the following. Suppose that, on the ambient space M, we have a non-trivial parallel spinor ψ0. This is the case whenM is Euclidean, but it is not the only case. Spin manifolds having non-trivial parallel spinor fields were classified, among others, by Wang [Wa] in 1989 and he saw that they include Calabi-Yau manifolds, hyper-K¨ahler manifolds and other 8- and 7-dimensional Riemannian manifolds. Then its restriction to the hypersurface (that I am denoting with the same symbol) satisfies
D/ ψ0= n 2Hψ0.
This means thatψ0 is an eigenspinor if the mean curvatureH of the hypersurface is constant. In the general case, we can use this spinorψ0onM as atest spinorin the variational characterization of the lowest eigenvalues and we obtain
λ21≤ Z
M
|D/ ψ0|2dM Z
M
|ψ0|2dM
=³n 2
´2 Z
M
H2dM vol(M) ,
whereλ1is the eigenvalue of the intrinsic Dirac operator with lowest absolute value.
This inequality was originally showed by B¨ar [B¨a] in 1998 and you can see that it is the analogue for the Dirac operator to the inequality for the Laplacian obtained in 1976 by Bleecker and Weiner that I mentioned at the beginning.
In the case of the Laplacian the equality was attained only by the round spheres.
Instead, it is an open problem to know when the equality holds in the B¨ar inequality.
We know that this equality implies that the test spinor field ψ0 is an eigenspinor for D/2 and we can easily see that, in this case, the hypersurface has constant mean curvature. So, the restriction of the parallel spinor field ψ0 is an eigenspinor corresponding to the eigenvalue (n/2)H. So an interesting question is: Does it occur for the Dirac operator the same thing as for the Laplacian one? That is, is the equality in the B¨ar inequality attained only by the spheres? If the answer is not, what can we say about constant mean curvature compact hypersurfaces of Euclidean space immersed by the first eigenspinors? This question should be the spinorial version of the problem posed by Yau and Ogiue for the Laplacian of minimal hypersurfaces of the sphere.
7 A lower estimate on bounding hypersurfaces
Now I want to talk about some lower bounds that we have found for the eigen- values of the intrinsic Dirac operator on compact hypersurfaces of Riemannian spin manifolds and about some geometric applications of them. Our estimates work only for hypersurfaces M which bound a compact domain Ω in the ambient spin mani- foldM. For example, this happens whenM is embedded and the ambient manifold is the Euclidean space or, in general, a manifold with vanishing first Betti number.
So, from now on we assume that our oriented hypersurfaceM is the boundary of a compact manifold with boundary Ω.
In this situation, our first tool is an integral version of the well-known Schr¨o- dinger–Lichnerowicz formula [Fr2, Li, LM]
D2ψ=−trace∇2ψ+1 4R ψ
which is valid for any spinor field ψ on Ω and where R is the scalar curvature of the ambient manifold. This integral formula [HMZ1]
Z
Ω
µ1
4R|ψ|2+|∇ψ|2− |Dψ|2
¶
= Z
M
³
hD/ ψ, ψi −n 2H|ψ|2´
has a difference with respect to the case of compact manifolds without boundary, where you have a zero on the right side: there is a boundary term where, after a suitable manipulation, the Dirac operatorD/ and the mean curvature functionH of the hypersurface appear. On the left side we use the Schwarz inequality and then we get the following integral inequality
Z
Ω
µ1
4R|ψ|2− n
n+ 1|Dψ|2
¶
≤ Z
M
³
hD/ ψ, ψi −n 2H|ψ|2´
where only the geometries of the domain and of the boundary and the two Dirac operators remain.
We can look at this integral inequality in two different ways. The first one is to emphasize the left side. From this point of view, it is useful to get information on the Dirac operatorDof the ambient manifold subjected to suitable boundary conditions [BW, FS]. The study of the natural boundary conditions that you can impose in order to solve first order equations of Dirac type was started by Lopatinsky [Lo] and Shapiro in the fifties, continued with the theory of pseudo-differential operators by Caldern [H¨o] in the sixties and systematized later by Seely [Se]. They gave precise standards to know when a given boundary condition for the Dirac operator on the domain Ω is elliptic and, as a consequence, the corresponding boundary problem is of Fredholm type and hence we have existence and regularity for its solutions.
Using this machinery, we have studied [HMZ2, HMR1] the spectrum of D under four different boundary conditions: the global Atiyah-Patodi-Singer (APS) [APS]
condition associated to the spectral resolution of the intrinsic Dirac operator on the boundary hypersurface; the local condition associated to a chirality (CHI) operator on the manifold (for example, when its dimension is even or when it is a space-like hypersurface in a Lorentz manifold) [FS]; the Riemannian version of the so-called (local) MIT bag condition [CJJT, CJJTW, J]; and finally a new global boundary condition obtained by modifying the APS condition (mAPS). We have shown that these four conditions satisfy ellipticity criteria and the corresponding boundary problems arewell-posedin the sense of Seely. Also we have found, under these four boundary conditions, the same lower estimate in terms of the scalar curvature that Friedrich discovered for the closed case, provided that the mean curvature of the boundary hypersurface is non-negative. Interestingly enough, the four conditions behave in differents ways with respect to the equality and our main motivation to study them was just to seeing that this equality is never achieved for the APS and MIT conditions. Instead the equality occurs for the CHI boundary condition if and only if the manifold is a half-sphere and it is attained for the mAPS condition if and only if the manifold supports a non-trivial real Killing spinor field and the boundary is a minimal hypersurface (this was a principal reason to look for and introduce this new boundary condition). For example, all the domains in a sphere enclosed by embedded minimal hypersurfaces have the same first eigenvalue for the Dirac operator subjected to that mAPS condition.
But there is a second way to approach to the integral inequality above. It is a question of taking out information about the geometry of the boundary hypersurface
M and not about the enclosed domain. To do this, we start with a given spinorφ on the boundaryM, for example, an eigenspinor for the intrinsic Dirac operatorD/, and use as a second tool the solutionψon the bounded domain Ω to the boundary
problem ½
Dψ= 0 on Ω
Bψ|M =Bφ along∂Ω =M,
where B is one of the elliptic boundary conditions above. We know that, in fact, there is a smooth solution to this problem because we have proved before that 0 does not belong in the spectrum ofDsubjected to the chosen boundary condition.
This technique consisting in putting in the boundary version of the Lichnerowicz integral formula the solution of a boundary problem on the domain determined by a hypersurface in order to get information on the hypersurface itself was the trick used by Witten [PT, Wi] in 1982 to give his spinorial simple proof of the positive mass theorem due to Schoen and Yau, although in 1977 Reilly [Re1] had already used it to study eigenvalues of the Laplacian. After them, this technique has been used by many authors, for example, by Min-Oo [Mi] to obtain a rigidity theorem for asymptotically hyperbolic manifolds, by Ros [Ro] to characterize the spheres as the only compact hypersurfaces embedded in the Euclidean space with constant scalar curvature and by Marc Herzlich [He] in his approach to the Penrose conjecture.
Now, as a third tool, we can make use of a property of the Dirac operator which has not an analogue for the Laplacian: a certain conformal covariance discovered by Hitchin [Hit]. This can be done by writing the (let me to call it) Reilly integral inequality above and solving the boundary problem that we have spoken about, not for the original metric, but for a carefully chosen metric in the conformal class and a convenient boundary condition. In fact, we have used the MIT boundary condition which is invariant under conformal changes of the metric and taken a conformal metric
h, i?=fn−14 h, i,
such that the change of scalef is the first eigenfunction of the following eigenvalue scalar problem
Lf =− 4n
n−1∆f+Rf= 0 Bf =− 2 n−1
∂f
∂N +Hf =νf.
The operatorL is the well-known conformal Laplacian and the boundary operator B could be called the conformal normal derivative along the boundary. This eigen- value problem for the conformal Laplacian was introduced by Escobar in 1992 [Es1]
because it naturally appears when one studies the Yamabe problem for manifolds with boundary (see also [Es2]). The corresponding first eigenvalue ν1 is not nec- essarily finite, although reasonable geometric assumptions on the domain Ω, such that non-negative scalar curvature, imply it finiteness. In this case, Escobar proved that the sign ofν1is invariant under conformal changes of the metric on the domain and that the associated eigenfunction f is positive. Then the new metric h, i? is
a scalar-flat metric on Ω and the boundaryM has positive, zero or negative mean curvature function according toν1 is positive, zero or negative.
8 Results
1) Combining these three ingredients we have obtained a sharp lower bound for the absolute value of the eigenvalues of the Dirac operator of a hypersurfaceM which bounds a compact domain in a Riemannian spin manifold. Precisely [HMZ3]
|λ| ≥ n 2ν1
where the equality holds only when the eigenspace associated to λ consists of re- strictions toM of spinor fields on the bounded domain which are parallel w.r.t. the conformally modified metric. A trivial use of the H¨older inequality implies that
|λ|vol(M)n1 ≥n
2Q(Ω, M),
where thisQ(Ω, M) is the relative Yamabe number of the domain enclosed by the hypersurface, which was studied also by Escobar [Es1] in 1994. Its value is the infimum over the scalar-flat metrics in the conformal class of the domain of the normalized total mean curvature of its boundary. If the equality holds here, then the eigenfunctionf associated toν1 must be constant and so the domain Ω carries non-trivial parallel spinor fields, the mean curvature of the boundary hypersurface is constant and the eigenspace corresponding to λconsists just of the restrictions of the parallel spinor fields of the domain.
2) Suppose now that the scalar curvature R of the ambient manifold is non negative. Then, from the variational characterization of the eigenvalueν1, we have that [HMZ1]
|λ| ≥ n 2 inf
M H
and the equality implies that H is constant and the space of eigenspinors for λ1 = (n/2)H is obtained by restricting to M the space of parallel spinors on the domain Ω. The equality is attained for any bounding hypersurface with constant mean curvature when the ambient Riemannian manifold carries a non-trivial paral- lel spinor. For example, besides spheres in Euclidean spaces, for any regular solution to the isoperimetric problem in a Calabi-Yau or hyper-K¨ahler manifold.
I would like to do one remark on this lower estimates. There are not similar estimates in terms of H, ν1 or Q(Ω, M) for the eigenvalues of the Laplacian. To see that, imagine a smooth embedded surface like this: a cylinder of radius one with ends closed by half-spheres of the same radius. This surface has (inner) mean curvature bounded by the mean curvature of the cylinder, that is, by 1/2 and this fact is independent of the length of the intermediate cylinder. However, when we increase the length of that cylinder the first non-null eigenvalue of the Laplacian becomes smaller and smaller.
3)A third consequence of our theorem is a simple spinorial proof of the Alexan- drov theorem [Al] for compact constant mean curvature hypersurfaces without self- intersections in the Euclidean space [HMZ1, HMZ3]. In fact, for such hypersurfaces the restrictions of the constant Euclidean spinors provide eigenspinors with eigen- vaule (n/2)H. As H >0 with respect to the inner normal, then our result implies that the first non-negative eigenvalue of the Dirac operator of the surface is (n/2)H and the corresponding eigenspace consists only of these type of restrictions. But, given such a constant spinorψ0, one can construct another spinor on the hypersur- face, taking Clifford product by the fieldHp+N
φ=γ¡
Hp+N¢ ψ0
and a direct computation implies that alsoφis an eigenspinor on the hypersurface corresponding to the same eigenvalue (n/2)H. Then it has to be restriction of a Euclidean constant spinor and so, ifuis vector tangent to the hypersurface,
0 =∇uφ=γ(Hu+ (dN)pu)ψ0.
Then we have that the differential of the Gauss mapNis proportional to the identity map, that is, the hypersurface is totally umbilical and so a hypersphere.
4) Finally, the last consequence of our lower estimate for the intrinsic Dirac operator on bounding hypersurfaces is a rigidity theorem for complete Ricci-flat spin manifolds [HM]. Suppose that such a manifold M admits an embedded hy- persurfaceM, which is isometric to ann-dimensional, sphere say of radius r >0.
You may suppose that the ambient space is 1-connected by lifting the embedding to the universal covering if necessary. The Gauss equation connecting the curvature tensors of the hypersurface and the ambient space gives, using its Ricci-flatness, that
S=n2H2− |σ|2
where σ is the second fundamental form. But S = n(n−1)/r2 because M is isometric to a radiusrsphere. Then the Schwarz inequality implies thatH2≥1/r2 and so the mean curvature does not vanish anywhere. Since M is 1-connected, M bounds two domains. We choose among them the domain Ω with positive inner mean curvature. ThenH≥1/r. A comparison theorem by Kasue [Ka] (a boundary version of Myers’ theorem) says us that Ω must be compact an so we can apply it our estimate. Then the first non-negative eigenvalue of the hypersurface Dirac operator satisfies
n
2r =λ≥ n 2inf
M H≥ n 2r
where the first equality comes from the fact that M is a sphere and the second one from our result. So we have the equality and then the eigenspace associated to this eigenvalue consists of restrictions of parallel spinors on the domain Ω. Hence one has a maximal number of parallel spinor fields on Ω and Ω has to be flat (see [Wa]). The analiticity of a Einstein metric [Be] implies that the ambient spaceM
is flat. So you will never find a round sphere embedded, for example, in a non-flat Calabi-Yau manifold.
I would like to finish by poiting out that, in joint works with my student A. Roldn [HMR2], we have obtained similar results in ambient spaces negatively curved.
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