Index and topology of minimal hypersurfaces in R^n
In this paper, we consider immersed two-sided minimal hypersurfaces in with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When , we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective Morse index bound by purely topological invariants, and is a generalization of [LW02]. Using our index estimates and ideas from the recent work of Chodosh-Ketover-Maximo [CKM15], we prove compactness and finiteness results of minimal hypersurfaces in with finite index.
Minimal hypersurfaces of the Euclidean spaces are critical points of the area functional. The Jacobi operator from second variation of area functional gives rise to the Morse index of the minimal surface. In Euclidean spaces , the second variation formula for a two-sided minimal hypersurface is given by
It induces a second order elliptic operator
where is the sum of square of principal curvatures, and is a compactly supported smooth function representing the normal variation. The Morse index of a compact subset is defined to be the number of negative eigenfunctions of with Dirichlet boundary condition. By the domain monotonicity of eigenvalues, when , . Hence we may define the Morse index of to be . This limit exists and may be infinity.
The classical Bernstein theorem [Ber27] asserts that an entire solution to the minimal surface equation in must be affine. Later, it was proved by Fischer-Colbrie-Schoen [FCS80], do Carmo-Peng [dCP79] and Pogorelov [Pog81] that the plane is the only stable (index ) minimal surface in . If we allow positive Morse index, there are lots of examples of complete immersed minimal surfaces in . In [Cos82], [HM85] and [HM90], the authors constructed embedded minimal surfaces of genus with any . The index of a genus Costa-Hoffman-Meeks surfaces is , by [Nay92] and [Mor09]. Another example of an immersed minimal surface with finite topology is the Jorge-Meeks surface [JM83]: for any integer , there is an immersed simply connected minimal surface with catenoidal ends. The index of a Jorge-Meeks surface with ends is [MR91]. These examples indicate a good control of the topology of a minimal surface in by its Morse index.
The relationship between the topology and Morse index of a minimal surface has been studied by many authors. From the work of Fischer-Colbrie [FC85], we know that if minimal hypersurface in a dimensional manifold has finite Morse index, then outside a compact part the surface is stable. [Cho90], [Ros06] and [CM14] prove that the index of a minimal surface in is bounded from below by a linear function of the number of ends and the genus. [CM14] also summarized various known results connecting the index and topology of minimal surfaces in with finite total curvature.
Similar study for minimal hypersurfaces in higher dimension has been limited, due to the lack of concrete examples and the inavailability of complex analytic tools. Examples of minimal hypersurfaces have been constructed in, for instances, [Cho] and [CH16], but none of which is complete and has finite total curvature. To our best knowledge, the only examples of complete minimal hypersurfaces with finite total curvature are the plane and the catenoid. In [CSZ97], Cao, Shen and Zhu proved that for all , complete two-sided stable minimal hypersurfaces have at most one end. Later Shen and Zhu [SZ98] proved that any complete stable minimal hypersurface in with finite total curvature must be a plane. For minimal hypersurfaces with positive Morse index, Tam and Zhou [TZ09] showed that the high dimensional catenoid has index . R. Schoen proved in [Sch83] that the catenoid is the only connected minimal hypersurface with two regular ends. Li and Wang [LW02] proved that finite index implies finitely many of ends. However, their result did not give an explicit control of the number of ends by the index of the minimal hypersurface.
It was pointed out by [SY76] that the existence of an harmonic -form violates stability. This was utilized by Cao-Shen-Zhu in [CSZ97] and by Li-Wang in [LW02]. Later Mei and Xu in [MX01] pointed out that if the minimal hypersurface has ends, then there exists a dimensional space of harmonic -forms. [Tan96] also investigated the connection between harmonic -forms and the stability in low dimensions.
In this paper, we combine an idea of Savo [Sav10] with the harmonic -form technique discussed above to get an effective estimate of certain topological invariants and the index of minimal hypersurfaces in . In fact, we can prove:
Let be a complete connected two-sided minimal hypersurface in , . Suppose that has finite total curvature, that is, is finite. Then we have
where is the dimension of the space of solutions of the Jacobi operator, and is the first Betti number of the compactification of .
By [Tys89], when and has Euclidean volume growth (that is, ), then has finite total curvature if and only if it has finite index. Therefore
Let be a complete connected two-sided minimal hypersurface in , . Suppose that has Euclidean volume growth. Then
Through a careful study of the rigidity case, we are able to get rid of the nullity term with the extra assumption that the dimension , or that there exists one point on where the principal curvatures are all distinct. Namely, we have
Let be a complete connected two-sided minimal hypersurface with finite total curvature. Suppose that , or that there exists a point on where all the principal curvatures are distinct. Then
The assumptions of Euclidean volume growth or finite total curvature in the previous two theorems are natural. In [Tys89], Tysk proved that all minimal hypersurfaces with finite total curvature must be regular at infinity (see [Sch83] for the definition of regular at infinity). That is, at each end the surface is a graph over some plane of a function decaying like . This precise large scale behavior of each end enables us to perform a more precise analysis.
Our theorem has some interesting applications in the study of minimal hypersurfaces in Euclidean space. For example, complete minimal hypersurfaces of index one is one of the most natural objects occured in geometric variational problems. In [LR89], López and Ros proved that the catenoid and the Enneper surface are the only index one complete connected two-sided immersed minimal surface in . It is unknown if the catenoid is the only index complete embedded minimal hypersurface in the Euclidean space. By [Sch83], we know that if a minimal hypersurface has two regular ends, then it is a catenoid. Theorem 1.3 is not strong enough to conclude this. However, we do have the following properties of the space of index minimal hypersurfaces in .
The space of complete connected embedded two-sided index minimal hypersurfaces with Euclidean volume growth, normalized such that , is compact in the smooth topology.
There exists a constant such that the following holds: for any complete connected embedded two-sided minimal hypersurface with finite total curvature and index , normalized so that , is a union of minimal graphs in .
Such property is not expected for a family of minimal hypersurfaces with larger index bound, as illustrated by the following example.
Let be the genus Costa-Hoffman-Meeks surface with ends, one planar end and two catenoidal ends behaving like near infinity. It is known that there is a family of deformed surfaces with three catenoidal ends whose growth rate near infinity is approximately , with . The surface qualitatively looks like three surfaces, each with one catenoidal end, joined by three catenoidal necks. The curvature of the surface is maximized at the three catenoidal necks. Now if we normalize each to , with maximized at where is on one of the three necks, then other necks of drifts to infinity as goes to infinity. In particular, for any , there is which is not graphical outside . However, the family have uniformly bounded index.
The second application is the finiteness of diffeomorphism types of minimal hypersurfaces in with Euclidean volume growth and bounded index. Using theorem 1.3 and ideas from the recent work of Chodosh-Ketover-Maximo [CKM15], we are able to get the following:
There exists such that there are at most mutually non-diffeomorphic complete embedded minimal hypersurfaces in with Euclidean volume growth and .
It would be interesting to see in more generality how the index of a minimal hypersurface in depends on its topological invariants. It is conjectured that a similar statement as in Theorem 1.3 should hold for . Even in dimension , we believe that the inequality of Theorem 1.3 is not optimal. For example, it does not answer the question of whether the higher dimensional catenoid is the only minimal hypersurface in Euclidean space of index . These are interesting questions to investigate in future.
The author would like to express his most sincere gratitude to his advisors, Rick Schoen and Brian White, for bringing this question to his attention and for several enlightening discussions. He also wants to thank Robert Bryant and Jesse Madnick for their insights in the rigidity discussion, and David Hoffman for a careful description of the Costa-Hoffman-Meeks surfaces. Special thanks go to the referee for many illustrating suggestions.
2. Spectral properties of minimal hypersurface with finite total curvature
We start by revisiting the following classical result of Fischer-Colbrie.
Theorem 2.1 ([Fc85]).
Let be a complete two-sided minimal surface of index . Then there exist orthonormal eigenfunctions of the Jacobi operator associated to negative eigenvalues, such that for any compactly supported smooth function on that is orthogonal to , .
In [FC85], the above theorem is stated for minimal surfaces in -manifolds. However, the same proof generalizes for complete two-sided minimal hypersurfaces in without much difficulty.
Let us now recall the definition for a minimal hypersurface to be regular at infinity.
Definition 2.3 ([Sch83]).
Suppose . A minimal hypersurface is regular at infinity, if outside a compact set, each connected component of is the graph of a function over a hyperplane , such that for ,
where is some constant.
In order to perform a more careful rigidity analysis, we use the extra condition that the minimal hypersurface has finite total curvature. By a result of M. Anderson [And84], finite total curvature implies that is diffeomorphic to a compact manifold minus finitely many points (in one-to-one correspondence to its ends). In fact, we have:
Proposition 2.4 ([Tys89]).
Suppose , in is a complete immersed minimal hypersurface with finite total curvature. Then is regular at infinity.
For our purposes, we use the fact that if has finite total curvature, then is bounded on , and the induced metric on tends to the Euclidean metric near infinity in the sense.
Let be a complete minimal hypersurface with index that is regular at infinity, and let be orthonormal eigenfunctions with negative eigenvalue given by theorem 2.1. Then for any function that is orthogonal to , . Moreover if , then is a solution of .
We first observe that each is in fact in . Indeed, is a solution of . Since is regular at infinity, the operator is a uniformly elliptic operator, and is bounded. Therefore by a covering argument and elliptic estimates, we have .
The first statement follows from a standard cutoff argument. Now let us assume . We will prove for any .
Let’s first assume is a compactly supported smooth function that is orthogonal to . Take a large so that is contained in . Choose a cutoff function which is on and outside . Denote , where are properly chosen real numbers such that is orthonogal to . Since is orthogonal to , each is independent of . By theorem 2.1, we have for any real number . Now
Note that . Since for all , we conclude
Let . Then and , . Therefore we get , so .
Now if is not compactly supported but is still orthonogal to , then can be approximated by compactly supported smooth functions that are orthogonal to . This implies .
Next we show . We use the fact that each is in , so it is a weak limit of a sequence of eigenfunctions of on . The statement now follows from a cutoff argument similar to the one before. Hence for in the span of and in its orthogonal complement, therefore for each in . ∎
3. The space of bounded harmonic functions on
The statement in this section can be found in [CSZ97] and [MX01]. We include the proof here because bounded harmonic functions are essential in the construction of test functions for the stability operator.
Let and be a complete minimal hypersurface in with ends. Then the are linearly independent bounded harmonic functions with finite Dirichlet energy.
When the constant function is harmonic. Suppose . Suppose for some compact domain , , where are the ends. For large enough, has boundary components. Solve the Dirichlet problem
By the maximum principle in . Using Schauder theory we get a uniform bound on for each compact . Therefore we may use Arzela-Ascoli to get a subsequence converging to in (). For , the function can be extended with constant value to a function on . Since harmonic functions minimize Dirichlet energy, . Therefore the function is a bounded harmonic function with finite Dirichlet energy.
Next we prove is not a constant function. Suppose the contrary. The function is in , hence by the Michael-Simon Sobolev inequality [MS73], we have for ,
Therefore is identically or . Without loss of generality we assume (otherwise consider instead). Choose some . Now take any smooth function which is identically on , on all other ends. Then is compactly supported. By the Michael-Simon Sobolev inequality and the fact that is compactly supported,
Letting we see that , contradicting the fact that the -th end has infinite volume.
By similar reasoning, the functions are linearly independent. Otherwise we would have . However, is the limit of some compactly supported harmonic functions taking as boundary values. An argument similar to the one before shows that such a cannot be constant. ∎
As a technical remark, in [CM14], the authors utilized the existence of harmonic -forms in a similar way. The major difference is that, harmonic functions on tending to constant on each end have finite Dirichlet energy if and only if . Therefore in [CM14] the authors have to use a weighted space rather than .
4. Proof of the main theorem
In this section we prove Theorem 1.1. We start by collecting a family of harmonic -forms.
Let be a complete minimal hypersurface in that is regular at infinity. Suppose has ends. Then there are linearly independent closed harmonic -forms on with finite Dirichlet energy.
Take the functions constructed in section 3. Their differentials are harmonic since and commute. We prove that is dimensional. The function is the limit of a sequence of harmonic functions with boundary values . By the maximum principle, each harmonic function in the sequence is identically . So is also the constant function . We see . Suppose are linearly dependent for some . Then . Therefore is a constant function on . Combine , we get a nontrivial linear combination of that equals , contradicting the linear independence of .
If , then we have linearly independent closed non-exact harmonic -forms . Then the set is a set of linearly independent closed harmonic -forms on . ∎
Now let us fix some notations. For any minimal hypersurface in , let be the Euclidean connection on and be the Levi-Civita connection of the induced metric on . Denote the Hodge Laplacian on -forms by . Suppose is two-sided with a unit normal vector . Take two vector fields on . Let be the shape operator defined by , and let be the second fundamental form defined by . For two parallel vectors in , let be their projection on . Let be a harmonic -form on and its dual vector field. With these notations, we have
Lemma 4.2 (Lemma 2.4 of [Sav10]).
, where is a tensor, defined by
to are standard facts about minimal hypersurfaces in Euclidean spaces. To prove , take to be the dual -form of the vector field . Then and are both harmonic -forms. Denote the connection Laplacian by . We have
By the Bochner identity for the harmonic -forms and , we have , . Therefore . Using the Gauss equation for in , we see that . Also by , . So . This proves .
To prove , we see that
We use and to simplify the first two terms. For the third term, we have
where the first equality is true by , and the third equality by , the fourth equality by the fact that is symmetric.
Using the above equality and , we get . ∎
Now we are ready to prove Theorem 1.1. Take to be the standard coordinates of . The vector fields are parallel vector fields in . Their projections onto are denoted by . Define the vector fields . For a harmonic -form on dual to a vector field , define the functions for . It is clear that is in . By lemma 4.2,
Regarding the integrability of , we have the following
defined above is in .
Fix an and a pair of as above. Note that is an one form. For the simplicity of notations, let us denote by and by . Since , we know that . Also since is bounded on , . Fix a point . Take a cutoff function which is supported in the Euclidean ball and is identically in , and satisfies . Since , we have
Integrate by parts and using the fact that is zero on , we have that
By Cauchy-Schwartz, we have
Let be a small number chosen later and using arithmatic-geometric mean inequality, we deduce that
Let be the cube inscribed to . Then is contained in , where is the cube with the same center under homothety by a factor of . Then
Now choose . Cover by parallel cubes with no common interior and side length same as , and also consider the -scale homothety of each cube in the covering. We deduce that each point is covered by at most times. Add the above inequality for each cube in this covering, we then get
hence . ∎
Suppose is finite. By Proposition 2.5, there exist smooth eigenfunctions of the Jacobi operator . Consider the linear system on
where run through .
Denote by the dimension of the space of harmonic -forms on . Now the linear system (4.2) has equations. If then there exists at least linearly independent harmonic -forms for which (4.2) is satisfied by , for each pair of with . For each such , by proposition 2.5, for each pair of . On the other hand,
For the first summand,
Therefore each is equal to zero. By proposition 2.5, is in the kernel of Jacobi operator. To conclude the proof of Theorem 1.1, we prove the linearly independent harmonic -forms generate at least linearly independent functions . Then . In fact, we have:
Let be an dimensional subspace of harmonic -forms on . Then the set has at least linearly independent smooth functions on .
Define a map , . We will prove that is injective. Suppose is a harmonic -form such that . That is, or for each pair . Then for some constant . Since is an orthonormal basis for , for each parallel vector field in and its projection on . In particular, at a point , choose and be a basis for , we get and .
Denote by the projection of onto the -th component. Since , at least one pair satisfies
For this particular , the space of functions spanned by are at least dimensional. ∎
Let us look closer at the equality case in the proof of Theorem 1.1. For any harmonic -form with , , in the kernel of Jacobi operator, we have . Locally, every can be written as for some smooth harmonic function . Then is equivalent to . Since is a basis for , we conclude that for every pair of tangent vectors . Now taking a local orthonormal frame of principal vectors on , we see that the above condition is equivalent to being diagonalized by principal vectors of . We are able to bound the dimension of the space of such functions when there is a point on where all principal curvatures are distinct.
5. Rigidity case
We prove that when in satisfies that there is a point where all principal curvatures are distinct, the space of harmonic -forms on satisfying , for each pair of , is at most dimensional.
Let be a connected minimal submanifold of an analytic manifold . Suppose that at one point on , all the principal curvatures of are different. Then the dimension of the function space
is at most .
Take an orthonormal frame in a small neighborhood of the point consisting of principal vectors with corresponding principal curvatures (all distinct), respectively. Then for any function with , letting and for , we get . Also implies . Now is an analytic manifold since it is a minimal hypersurface of an analytic manifold. By the unique extension theorem, any harmonic function is uniquely determined by all its derivatives at one point . We prove that if a harmonic function satisfies the extra condition that commutes with the shape operator , all the covariant derivatives are uniquely determined by , so the dimension of all such functions is at most .
Let us prove that if and then all derivatives . We’ll proceed by induction on . The cases of are given as assumptions. Now suppose , and that any covariant derivatives of with order less than or equal to are zero. Consider a covariant derivative . We separate two cases.
Not all of ’s are equal. Then after switching the order of taking derivatives finitely many times, we will get an expression of with . Every time we switch two consecutive indices , the difference we get is a curvature term depending linearly on lower order derivatives of at . By assumption all lower order derivatives of at are zero. On the other hand, since , . Therefore, in this case, .
are all equal. Without loss of generality we may assume . Since , . Therefore