This is a sequel to my previous post Reilly type formula and its applications. (Oops… it’s been a long time!)
Integral formulas have always been an important tool for studying various analytical and geometric problems on Riemannian manifolds. The Reilly’s formula [R1] and the Hsiung-Minkowski formulas [H] are typical examples that have yielded many classical results. Often, these formulas produce some integral identities or inequalities where the vanishing of the integrand produces useful geometric consequences. Despite the simplicity of such an idea, this method achieves many results. Let us for example mention the work of Ros[Ros] who has combined the Reilly and Minkowski formulas to show that a compact embedded hypersurface in with one of the higher order mean curvatures being constant is a sphere. On the analytic side, it is well-known that the first non-zero eigenvalue of the Laplacian with respect to various boundary conditions gives very important analytic information of a Riemannian manifold. Integral formulas is one of the main tools in obtaining various eigenvalue estimates. For example, Choi and Wang [CW] used the Reilly’s formula to prove that for a compact orientable minimal hypersurface of a compact orientable with Ricci curvature bounded from above by , the first nonzero eigenvalue of satisfies , this is closely related to a conjecture of Yau [Yau].
In this note, we apply some integral formulas to obtain some geometric and analytic results. In the first part, using conformal vector fields on space forms and applying Reilly’s and Hsiung-Minkowski formulas, we obtain some integral inequalities on a convex hypersurface in space forms, and discuss its sharpness. They can be regarded as some rigidity results in space forms. Here is an example of our results, in which we show a characterization of a convex hypersurface in space forms, which is a particular case of Theorem 10 and Theorem 11:
Theorem 1 Let be a compact convex hypersurface of which encloses . Suppose is circumscribed by the geodesic sphere centered at . Then
where is the normalized -th mean curvature of and . The equality holds if and only if is . There are analogous results for and . For example, if is a convex hypersurface contained in the open hemisphere centered at and , then
Here is the distance from . The equality holds if and only if is .
This generalizes Garay’s result [Garay] that if a convex hypersurface in has mean curvature , then is the sphere of radius .
In the second part of this note, we will apply Reilly’s formula to obtain some lower bounds for the first nonzero eigenvalue of the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold whose Ricci curvature is bounded from below. Our main results, Theorem 15 and 17, are natural extensions of the results of Escobar [E1],Xia[X], Wang-Xia [WX] and Raulot-Savo [RS]. For example, in Theorem 17 we extend the results of[X] and Raulot-Savo [RS]:
Theorem 2 Let be a compact orientable Riemannian manifold with boundary . Suppose the Bochner curvature or on is bounded from below by . Assume that the lowest -curvature of is nonnegative, where . Then for , we have
where (resp. ) is the first nonzero eigenvalue of the Hodge Laplacian on the exact (resp. co-exact) -forms on . The equality can hold only when , with the -curvatures and the -curvatures being positive constants. If, furthermore, has non-negative Ricci curvature, then the equality holds if and only if is isometric to a Euclidean ball. The condition on Ricci curvature can be removed if or .
The notions and will be explained in Section 4. Let us just mention that when , is the minimum eigenvalue of the second fundamental form of and is the minimum of its mean curvature, is just the Ricci curvature and is the first nonzero eigenvalue of the Laplacian on functions on .
We will also give a sharp lower bound of in terms of the first nonzero Steklov eigenvalues for differential forms, as well as some lower and upper bounds for the Steklov eigenvalues in terms of (Theorem 17). It is also interesting to see that when , a simple extension of the result of Hang-Wang[HW] gives an improvement of Choi-Wang’s result mentioned above. Indeed, we can prove that and the estimate is sharp (Theorem 21). It may have some independent interest.
In Section 2, we set up the notations and introduce the Reilly and Hsiung-Minkowski formulas. In Section 3, we establish some integral inequalities of convex hypersurfaces in space forms. In Section 4, we prove the various estimates for the Hodge Laplacian eigenvalues and also Steklov eigenvalues for differential forms on a manifold with boundary.
2. Reilly’s formula and Hsiung-Minkowski formulas
Let us first set up the notations. Throughout this note, will denote an -dimensional connected oriented Riemannian manifold () (with or without boundary). If is without boundary then we will always denote a closed hypersurface on by , otherwise we assume is compact has a smooth compact boundary . We will denote the Levi-Civita connection on and by and respectively. The Laplacian on and will be denoted by and respectively and let be the Ricci curvature of . We define to be the unit outward normal on w.r.t. and and to be the second fundamental form and the mean curvature of respectively.
Reilly’s formula states that:
Theorem 3 [R1] Let be a smooth function on , and . Then
To state the Hsiung-Minkowski formulas, we define
Definition 4 A vector field on is conformal if there is a smooth function on such that
As , it is easy to see that if is conformal.
Next, we define the -th mean curvature of in as follows. Suppose are the principal curvatures of (eigenvalues of the second fundamental form ), we define by the identity
We define the normalized -th mean curvatures to be .
We are now ready to state the Hsiung-Minkowski formulas:
Theorem 5 [H][R2] Suppose is a space form. Let be a conformal vector field on and be an unit normal vector field of . Then
for . For , this holds with no assumption on being a space form.
Proof: Let us illustrate the proof for for example. Let be the vector field on defined by where is a local orthonormal field on with , then
The result follows by applying divergence theorem.
3. Some integral inequalities in space forms
In this section, we first study the conformal vector fields on space forms. We then apply Reilly’s and Hsiung-Minkowski formulas to deduce some sharp integral inequalities for convex hypersurfaces in space forms.
By rescaling we can assume the curvature of a space form is . We will denote the space forms
will give a conformal vector field on . In fact, if then
From (3), we have
Lemma 6 On , if is chosen by (2), then
Using (3), by direct calculations, we have
Lemma 7 On , for chosen by (2), we have
Lemma 8 Suppose is a region in which has smooth boundary , if is chosen as in (2), then
where and .
Proof: We have . Let be an orthonormal basis of such that . Then
The result follows from (3).
Lemma 9 With the notations in Theorem 3, suppose is a region in which has smooth boundary , if is chosen as in (2), then
Proof: Recall and . By the formula , we compute
So by divergence theorem,
We now state our first main result:
On the other hand, by Lemma 9,
Thus the inequalities in all three cases hold.
Suppose the equality holds, since , from (6), we have if . In the case where , we have and . From this we can easily deduce that is the equator, i.e. the geodesic sphere of radius .
Remark 1 We remark that when , there is no loss of generality to assume that is contained in a closed hemisphere, since by the result of [DW], is either totally geodesic (i.e. a great hypersphere) or is contained in an open hemisphere.
Using similar techniques, we have
Proof: Recall . Lemma 9 implies
On the other hand, since , we have . So
We will prove the remaining cases by induction, and let us do it only on , since the other two cases are similar. The induction step is done by observing
which we have used (5).
Suppose any of the equalities holds, then and hence , i.e. it is the sphere centered at with radius .
Remark 2 Recall that . When , by Theorem 11 we have . This recovers a result due to O.Garay [Garay] that if is convex and
then is the sphere of radius . The condition can also be replaced by .
To state the next result, we need to have a notion of the center of mass of in . Let be a fixed point on . We use the following models for :
with metric induced from
With this understood, we have
Lemma 12 Suppose is a compact hypersurface in whose center of mass is , then its first non-zero eigenvalue of satisfies
where is the tangential component of onto .
Proof: By min-max principle [Yau], we have if . In particular,
Theorem 13 For a compact convex hypersurface of , assume that its center of mass is if and it is inside the open hemisphere centered at if , then
where and . The equality holds if and only if is a geodesic sphere.
Proof: Note that and are independent of the center of mass, so we can assume that in all cases, its center of mass is . By Lemma 9,
On the other hand, by Lemma 12, . Therefore
Remark 3 In the case where , Theorem 13 implies that if is convex and , then is a sphere. This is also a result of O. Garay [Garay].
4. Applications to eigenvalue estimates
In this section, we will prove several lower bounds of the first nonzero eigenvalue of the Hodge Laplacian on differential forms on the boundary of a Riemannian manifold . These results are the natural generalizations of some results in [E1],[HW], [IM], [RS], [X] and [X].
First we set up some notations. Fix and let be the principal curvatures of at w.r.t. the outward unit normal . We define the -curvatures (not to be confused with the -th mean curvature) to be all the possible sums where . We can assume , then we define the lowest -curvature to be
We also define
Note that the second fundamental form is bounded from below by and is the mean curvature. It is easy to see that if , . and that implies .
We denote by and to be the exterior derivative and its adjoint w.r.t. the inner product on respectively. The Hodge Laplacian of a -form on is defined by
for . Our sign is chosen such that is the second derivative for functions on . Recall the Bochner formula (see e.g.[P] p.218 Theorem 50):
where is a self-adjoint endomorphism on , which is determined by the Riemann curvature tensor on . This term is sometimes called the Bochner curvature term. When , and by [GM], where is the lowest eigenvalue of the curvature operator on . However, is usually much weaker than the curvature operator being nonnegative.
We define the shape operator on and define by
We also define to be zero. For example, if is a 1-form, then . Observe that and that the eigenvalues of are exactly the -curvatures of , therefore
We define (respectively ) to be the nonzero eigenvalue for the exact (respectively co-exact) -forms on . By Hodge decomposition theorem and Hodge duality (e.g. [Wa]), we have
From this we see that to determine , it suffices to determine for .
The following formula is the generalization of Reilly’s formula to differential forms.
Theorem 14 ([RS] Theorem 3) Let , , then
where the boundary term is given by
Here is the inclusion and is the Hodge star operator on . We will also denote by and the exterior derivative and its adjoint on respectively.
The classical Reilly’s formula (Theorem 3) can be recovered by setting .
We now state our first main result in this section.
Proof: Note that by Hodge decomposition theorem and Hodge duality, and by (14) below, both and are bounded from below by .
Let be an co-exact eigenform on with eigenvalue , i.e. . Then is an exact -eigenform with eigenvalue . By Theorem 2 of [D] (p.148), there exists an -form on such that and on . Let . Then
Using Reilly’s formula on ,
The condition implies . From the above, as , this shows that , which proves (10) in the case where . So in the following we can assume .
As and , the inequality (11) becomes
As is not identically zero, we conclude that
where . This implies either
In view of (11), we conclude that the second case holds. i.e.
Suppose the equality holds, then from (11), . As is parallel, is constant, and as , this constant is nonzero, which we can assume to be . The curvature term is given by (see e.g.[P] p.218 Theorem 50):
where is a local orthonormal frame on , is its dual frame and is the curvature operator on . Here is the Clifford multiplication on a differential form . Since , we have and so . Therefore from (11)
So we now have . Therefore from (12),
Now we suppose, furthermore, that . As ,
On the other hand, by Stokes theorem,
From these we have
Recall that we have for , so . Thus
By[Ros] Theorem 1, as , we conclude that is isometric to a Euclidean ball.
Using and , we have, by (13),
As and , so the condition is redundant for or . Finally, it is well-known that (see e.g. [GM])
From this it is easy to see that the equality holds on any Euclidean ball, with .
To state our next result, we need to define the Steklov eigenvalues, as follows. Let , . Then there exists a unique -form such that (see e.g. [Sc] Theorem 3.4.6)
We define the Steklov operator by
By [R2] Theorem 11, is an elliptic nonnegative self-adjoint pseudo-differential operator of order one. Thus the eigenvalue problem
has a discrete spectrum
We will write for . Here we use the convention in [R2] that is the smallest nonnegative eigenvalue of . Thus in the classical case where , i.e. for , being the unique harmonic extension of to and
the first nonnegative eigenvalue of is zero, corresponding to the constant functions on . So in our convention, and is the smallest positive eigenvalue, usually called the first Steklov eigenvalue of . We will simply denote by .
We remark that the first eigenvalue of satisfies the min-max principle ([R2] Theorem 11):
When , we also have the following min-max principle for the smallest nonzero Steklov eigenvalue (see for example [Hen] p.113):
Proof: For all we can assume that the center of mass is , i.e. for . By (17),
Multiply the above by , we have, by Cauchy-Schwarz inequality,
The inequality follows by applying the Hsiung-Minkowski formula (Equation (5)).
Suppose the equality holds and is not identically zero, then are Steklov eigenfunctions associated to , thus by Lemma 6, . But then for any tangential vector of , we have . Thus is constant and is a geodesic ball of radius , where .
Remark 5 By the Hodge-deRham theorem for manifolds with boundary ([Sc] Theorem 2.6.1), any cohomology class of the deRham cohomology space (with real coefficients) is uniquely represented by such that
We will denote the space of all such by . So from (16), we see that is positive if and only if . Therefore we are interested in only when .
By Hodge duality, the relative deRham cohomology space (cf. [Sc] p.103) is isomorphic to the vector space
called the space of Dirichlet harmonic fields.
Theorem 17 Let be a compact orientable Riemannian manifold with boundary . Let . We assume is nontrivial if (corresponding to ). Suppose the Bochner curvature of is bounded from below by , the -curvatures of are bounded from below by and . Let and let to be if and if . Then
- We have the following upper bound for :
- Assume , then we have the following lower bounds for and :
- Assume and . We have either
provided that it is well-defined. (If and , we will show that , see also Remark 6. )
- Assume , and . Then
If , the condition can be replaced by and .
- The inequalities (19) and (20) are actually strict (if ). Any of the equality cases in (18), (22) or (23) can hold only when , with the -curvatures and -curvatures both being positive constants.
Suppose has non-negative Ricci curvature. Then the equality in (18) or (22) holds if and only if or , and is isometric to a Euclidean ball. The condition on Ricci curvature can be removed if . The equality case in (23) can hold if and only if , and is a Euclidean ball.
Proof: Let be an co-exact -eigenform on with eigenvalue , i.e. . Then is an exact -eigenform on and by [Sc] Lemma 3.4.7, there exists an -form on such that
By Stokes theorem,
So we have . Let , then is a harmonic field, i.e. .
By the Friedrichs decomposition for harmonic fields , as is exact, there is a unique co-exact -form on such that (see [Sc] Theorem 2.4.8 and its proof):
Let , then as ,
Thus by (16),
As , we also have , so
This is true due to the min-max principle for (Equation (17)), together with the fact that and .
Squaring this inequality gives . From this we see that and (21) follows.
As is co-closed and , it is also co-exact. So by [R2] Proposition 14, , and (23) follows. If , and , then by [Sc] (Theorem 2.6.4, Corollary 2.6.2 and Theorem 2.6.1), , thus this later condition can be dropped.
If any inequality sign of the inequalities (18), (19), (20) or (22) becomes an equality sign, then one of the inequalities in (18) or (19) is an equality. Assume one of these holds. The inequalities (31) (or (34)) and (11) then become equations. So we have the -curvatures are constantly equal to , and the -curvatures are equal to the constant . In particular, .
Here is the Hodge star operator on and is a local orthonormal frame on . The last two formulas are standard and are included here just for convenience (e.g. [Sc]). For the first two formulas, see [RS] Section 2 and 6. Using (36), we compute
As , we conclude that , or
We can now proceed in exactly the same way as the proof of Theorem 15 to show that must be a Euclidean ball if , which we can w.l.o.g. assume to be the standard unit ball . But then by [RS3] Corollary 4,
As and by (15), we conclude that if , the equality in (18) or (22) holds if and only if . For , it is well-known that , from this we can also conclude that the equality in (18) or (22) holds if and only if is a Euclidean ball.
Suppose the equality in (23) holds, then by (31) or (34), and by the same reason as above, the -curvatures are constantly equal to , the -curvatures are constantly , and . In particular, in view of (23), so from (35), we have
In view of (23), we deduce that . We can then proceed as before to conclude that if , then is a Euclidean ball. But then by (15) and (37), the equality cannot be attained on a Euclidean ball if . If , then from (37) we see that the equality is attained if and only if , on a Euclidean ball.
- Escobar ([E2] Theorem 8) showed that if , then , so (22) is well-defined. Also, (23) is a generalization of [E] (Theorem 9) and [RS2] (Theorem 8, Theorem 9).
- Theorem 17 (17) is an extension of [X] Theorem 1.1, in which they provided an upper bound for which corresponds to our result when and .
- We suspect that (22) holds whenever and in this case we have , but we are unable to show it for the time being.
In [YY], Yang and Yau proved that for a compact Riemann surface of genus , for any metric on , . Combining this result with Theorem 15 and 17, we have several corollaries. Let us only state the following:
Corollary 18 If , then under the assumptions of Theorem 15, we have .
Example 1 Although the estimate of Theorem 15 is not sharp when , , by examining the case where is a geodesic circle of radius in a hemisphere (), it is found that the error is within . Indeed, in this case, , , we have . The error tends to zero as .
On the other hand, by [E2] Example 5, the first nonzero Steklov eigenvalue of the geodesic ball of radius in is computed to be . By direct computations, it is found that the error in Theorem 17 (17) is which is (very) slightly better than that of Theorem 15.
Corollary 19 Let be a compact orientable Riemannian manifold with boundary . Suppose the Ricci curvature of is nonnegative, for some , and is nonnegative, then is isometric to the unit ball in .
Corollary 20 Suppose be a compact surface with (not necessarily connected) boundary with the Gaussian curvature . If the geodesic curvature of satisfies , then its length . The equality holds if and only if is isometric to the Euclidean disk of radius .
Proof: By Gauss-Bonnet theorem, , thus has only one component. By Theorem 15, . The equality holds if and only if is a Euclidean disk of radius . As , the result follows.
In [CW], Choi and Wang proved that if is a compact orientable manifold whose Ricci curvature is bounded from below by and is an embedded orientable minimal hypersurface in , then . Since their proof are essentially the same as that of Theorem 15, their result can be improved slightly to . This is related to Yau’s conjecture [Yau]. It is easy to see that the coordinate functions are eigenfunctions of a minimal hypersurface of (whose Ricci curvature is ) with eigenvalue . Yau conjectured that the first eigenvalue is actually . Escobar also have a similar conjecture in [E1].
In the two-dimensional case, an embedded minimal submanifold is reduced to a simple closed geodesic, the result of Choi-Wang can be improved to , by a result of Toponogov Top on the length of a closed geodesic. More generally, we have the following result which is a simple extension of the result in[HW], which may have some independent interest:
Proof: By[HW] Theorem 3, we have . The equality holds if and only if is isometric to the disk , . Therefore if , is isometric to the standard hemisphere. But then , thus we can apply the same argument to to deduce that is . In general, if , then by Gauss-Bonnet theorem, as is a topological sphere,
So if , then on and so is .