A Compactness Theorem for Riemannian Manifolds with Boundary and Applications
In this paper we prove weak (and thus ) compactness for the class of uniformly mean-convex Riemannian -manifolds with boundary satisfying bounds on curvature quantities, diameter, and -volume of the boundary. We obtain two stability theorems from the compactness result. The first theorem applies to -manifolds (contained in the aforementioned class) that have Ricci curvature close to and whose boundaries are Gromov-Hausdorff close to a fixed metric on with positive curvature. Such manifolds are close to the region enclosed by a Weyl embedding of the fixed metric into . The second theorem shows that if a -manifold with has Ricci curvature close to (resp. , ) and mean curvature close to (resp. , ), then is close to a metric ball in the space form of constant curvature (resp , ).
Cheeger-Gromov compactness (, ) asserts that the class of Riemannian manifolds (of fixed dimension) satisfying , , and is precompact in the topology, any . Here , and respectively refer to the sectional curvature, diameter and volume. It is natural to ask if a result like this is true for certain classes of manifolds with boundary. Kodani () has proven an analogue of Gromov’s compactness theorem for manifolds with boundary. To describe the result, first consider the class of Riemannian -manifolds with boundary satisfying the conditions
Here is the second fundamental form of in .
If is allowed to be arbitrary, then this class is not even precompact in the topology.
However, Kodani shows that there exists , depending upon and ,
so that if , then any sequence in the class subconverges in the Lipschitz topology to a limiting
Riemannian metric. The precise control required
of is a definite restriction on the applicability of the theorem. Moreover, the regularity of the limiting metric is not optimal. Since
the mean curvature involves one derivative of the metric, one expects to gain control of the
first derivatives of and therefore obtain, for instance, an limit.
Extending techniques introduced by Anderson (), it is shown in  that the class of Riemannian -manifolds with boundary satisfying
is precompact in the weak topology. is the Zygmund space intermediate between and , is the
boundary injectivity radius defined in section 2 and is the Lipschitz norm.
Weak precompactness means that any sequence subconverges in the topology
(any ) to a limit.
The limit is necessary for the applications in , therefore
Lipschitz control of is natural in this context. We note that if one strengthens the control of
Theorem 1.1 from pointwise to Lipschitz, then the techniques in 
allow us to obtain
weak convergence as well. The details of this are minor however and we will not discuss them here (see
also the remarks below Theorem 1.1).
Let us assume throughout that all manifolds being considered are connected. Write for the class of compact Riemannian -manifolds with connected boundary satisfying
is precompact in the and weak topologies, for any and any . Consequently has only finitely many diffeomorphism types.
The sectional (and thus Ricci) curvature bounds give control of the metric in appropriate charts (e.g. harmonic coordinate charts) in the interior of .
Similar analysis applied to
the intrinsic boundary metric gives
control of in charts of uniform size on . This information can be used to gain
control of the tangential components of the metric in a neighborhood of the boundary (see the
proof of Theorem 1.1).
The nontangential components of the metric however are controlled by the mean curvature. Here one
only obtains control of these components in a neighborhood of the boundary, so that the regularity part of
Theorem 1.1 is limited by the pointwise control of . Therefore we could
convergence result by controlling in a stronger norm. If we assume for instance that
is controlled in an appropriate trace space (see  for an
example of an ‘appropriate trace space’),
then we obtain weak convergence in the statement of Theorem 1.1. We could
similarly assume that is bounded in the or Lipschitz topology (as in )
and respectively obtain a weak or weak convergence result. Since the proofs
of these results are similar to the proof of Theorem 1.1 (and because it is
more natural from a geometric standpoint to work with a pointwise bound on ) we will
leave the details to the interested reader.
We will use Theorem 1.1 to prove two ‘geometric stability theorems’ regarding -manifolds with boundary. Write for a smooth, closed, oriented surface with Gauss curvature . From the solution of the Weyl problem (cf. ), there exists a smooth isometric embedding whose image is unique up to rigid motion. Choose such an immersion and write for the convex solid region bounded by . Then is a smooth, flat manifold with boundary .
Suppose is a compact, oriented, simply connected Riemannian -manifold with connected boundary. Write for the induced metric on and write for the Gauss curvature of . Suppose that and . To every there exists a number so that if
then there exists a diffeomorphism , and
where is the standard Euclidean metric on and is the Gromov-Hausdorff distance.
Using the same techniques we can obtain a somewhat different result in the special case that is a ball. Write for the unit ball in .
Suppose is a compact oriented Riemannian -manifold with connected boundary and that . To every there exists so that if
then there exists a diffeomorphism , and
Theorem 1.2 can be viewed as a generalization of the well-known rigidity theorem of Cohn-Vossen (, ). Cohn-Vossen’s theorem, part of the solution of the Weyl problem mentioned above, states that an analytic immersion
of a closed surface with Gauss curvature has a unique image
modulo a rigid motion of . Pogorelov ()
removed the restriction on the regularity of the immersion. The result can be restated, via the developing map,
as a theorem about flat -manifolds with boundary. Thus if
and are compact, simply connected, flat -manifolds with isometric
boundaries that have positive Gauss curvature, then is diffeomorphic to and
the metrics , are in the same isometry class.
Such manifolds are therefore ‘geometrically rigid.’ Theorem 1.2 is a natural
generalization of Cohn-Vossen’s rigidity theorem in this context.
Let us provide another application of Theorem 1.1, motivated by Hopf’s rigidity theorem (). Hopf’s theorem states that the image of a isometric immersion
of a metric on with constant mean curvature is a (Euclidean) sphere. Essentially the same proof shows that the image of a isometric immersion
of a metric on into hyperbolic space is the boundary of a metric ball, provided . Here is the trace of the second fundamental form with respect to the outward normal, so that in our notation every distance sphere in has mean curvature . Almgren () has shown (making use of Hopf’s proof) that any analytic minimal immersion of into is congruent to the equator. To each of these rigidity theorems we associate a geometric stability theorem.
Let be a compact, oriented Riemannian -manifold with connected boundary and
suppose that . To every there exists such that
then there exists a diffeomorphism and
then there exists a diffeomorphism and
where is a metric ball in hyperbolic space with Gauss
curvature of the boundary equal to , and
is the standard metric of curvature .
then there exists a diffeomorphism and
where is the upper hemisphere in and is the standard metric of curvature .
In section 2 we provide definitions and well-known background results necessary for the proofs to follow.
The proof of Theorem 1.1 is contained in section 3,
the proofs of Theorem 1.2 and Corollary 1.1 are contained in section
4, and the proof of Theorem 1.3 is
in section 5.
Acknowledgement. The author would like to thank his thesis adviser Michael Anderson for suggesting the topic and for many valuable comments and conversations.
2. Background Results
Let us begin with a few observations about the elements of . Write and note that is smooth off of the cut locus of . Write
so that is the outward normal of . Write for the second fundamental form, thus . Here is the usual covariant derivative. The mean curvature of is the trace . If is a basis for such that , then the Gauss equation reads
This implies that each pair is uniformly
bounded. Together with the fact that is uniformly controlled we see that
each is uniformly controlled as well (in fact ).
A consequence of the uniform bound on is that the elements of have a uniform upper bound on the intrinsic diameter of the boundary (see [23, Thm 1.1]). Thus we will assume without loss of generality that .
Write for the focal locus distance of . Let us recall a basic result from comparison geometry.
Suppose and are any real numbers. Let be the smallest positive solution to
(a) If and , then .
(b) If and , then .
Write for the cut locus distance of . Write for the normal bundle of and define the normal exponential map
Define the boundary injectivity radius to be the largest such that is a diffeomorphism on . It is a standard fact that
Let us show, using the proof of [2, Lemma 2.4], that any has uniformly bounded below.
Suppose . Then
for as defined in Lemma 2.1. If in addition , then
Since , it is enough to show that if , then
Therefore suppose that . Choose an arclength parametrized geodesic realizing the cut locus distance. Thus
is a minimizing geodesic from to that is orthogonal to at the endpoints and such that . Consider the index form
where and are vector fields along and orthogonal to . Since there are no focal points along it follows that
for any such . Therefore choose an orthonormal basis for and define to be the parallel translation of along . Then
establishing the first assertion. If in addition , then
which contradicts the fact that . ∎
2.1. Convergence results
A sequence of Riemannian manifolds (with or without boundary) converges to in the topology, ,
and there exist diffeomorphisms so that in the topology on . is the
usual Sobolev space of functions (or tensors) with weak derivatives in . To be somewhat more precise about the
definition of convergence, we require that has an atlas of charts in which in
in each chart. Similar definitions hold for convergence in other function spaces.
For an introduction
to the convergence theory of Riemannian manifolds see for instance
, , ,,
It is useful to discuss convergence theory in terms that only refer to the local geometry of . For this purpose we use terminology first introduced by Anderson in . Given and a number , define the harmonic radius at , denoted , to be the largest number satisfying the following conditions. There exists a harmonic coordinate system (i.e. ) centered at and containing the geodesic -ball on which there holds, for each multi-index with ,
where is the standard Euclidean metric.
The conditions are invariant under simultaneous rescalings of the metric and
the coordinates, so that scales like a distance function, i.e. .
discussion we will assume that is fixed, say , so in particular there is no loss of clarity when supressing the dependence of on .
If , define to be the
largest such that and such that there exists a
harmonic coordinate system centered at , containing the geodesic -balls and
and satisfying equations (2.3)-(2.4). Here refers to the harmonic radius of in the manifold , and
is defined to be the ball of radius in . A harmonic coordinate
system at is defined to be a system of coordinates
so that form a harmonic coordinate system at in , on and each is
harmonic on . We then define the harmonic radius of , , to be the largest so that for
each in , either or there exists
with and .
It is clear how to extend the definition of harmonic radius to other function spaces. We could also consider for instance the harmonic radius, , . By the Sobolev embedding theorem, if then the harmonic radius controls the harmonic radius, . The harmonic radius of a manifold with boundary was previously defined and studied in , where it is shown that (for instance) if is a Riemannian manifold with , then admits an atlas of harmonic or boundary harmonic coordinate charts in which (see also ).
In order to focus our attention on the local geometry of near the boundary, we will also define the boundary harmonic radius . The boundary harmonic radius retains all of the important properties of the harmonic radius. For instance, it is continuous with respect to the topology, .
Suppose that in the topology, . Then
The corresponding result for the harmonic radius of a complete manifold without boundary is done in , and
the continuity of the boundary harmonic radius (under convergence) is done in . The proof for
the boundary harmonic radius is nearly identical to these cases, thus we will not describe it here.
The same method used in the proof of Lemma 2.3 also implies that for a fixed manifold the function is continuous.
The link between the harmonic radius and convergence is established by the following theorem (See for instance  and [20, Theorem 72]. The proofs there are about convergence of manifolds without boundary. However, the Banach-Alaoglu theorem allows the result to be extended to weak convergence, and we may extend the result to manifolds with boundary by simply including boundary harmonic coordinate charts in the analysis.)
Suppose is a sequence of Riemannian manifolds with boundary such that (for some ) the harmonic radius and . Then there exists a smooth manifold with boundary , , so that subconverges in the weak topology to . In particular, the harmonic coordinate charts on subconverge weakly in to harmonic coordinates on .
In the statement of Theorem 2.1 the diameter bound is used to obtain a uniform upper bound on the number of coordinate charts needed to cover . We could also remove the diameter bound and consider pointed convergence. Given a sequence of points we say that converges to in the topology if there exist real numbers , , and compact sets , so that
so that in and .
A similar definition of pointed convergence could also be used to formulate a local version of Theorem 2.1.
We will make use of the following result.
Let be a compact Riemannian -manifold with boundary with . Suppose that for each and each there holds
Then for any there exists a constant so that the harmonic radius satisfies
Proof of Theorem 2.2.
We will outline the proof of this well-known result. Arguing by contradiction, suppose there were a sequence satisfying the hypothesis of Theorem 2.2 but with
where . Consider then the rescaled sequence . The scaling behavior of the harmonic radius implies that (calculated with respect to ) . Since equation (2.5) is scale-invariant it follows that
Thus Theorem 2.1 implies that converges
in the pointed weak topology to a complete Riemannian manifold .
As we see that . In particular, in . This allows one to improve the convergence from weak to (strong) (see ). Then Lemma 2.3 implies that . However, the limit is a complete, flat, Riemannian manifold and is therefore isometric to a quotient of . The volume growth condition implies that (for all )
This implies that , in contradiction to the fact that . ∎
3. The Precompactness Theorem
The proof of Theorem 1.1 proceeds in two basic steps. The first is to
control from below the Euclidean volume growth of any point in the interior of . The second is to
the boundary harmonic radius is bounded from below. These two facts together with
imply that the harmonic radius is uniformly bounded below, so that
Theorem 2.1 establishes the result.
Let us begin by controlling the volume of small cylinders in with base . Therefore define, for ,
Suppose and choose . Suppose there exists so that . Then there exists a constant , depending only upon , so that
First note that volume comparison (in ) implies that
for some that only depends on , , and . Let be the level set
Since , it suffices to show that is uniformly controlled in terms of and . We note that if another constant is chosen so that the inequality
holds for all , then
This estimate is proved for instance in Lemma 3.2.2 of  (with a specific choice of ).
There exists a constant so that for each and each ,
Put . Choose an arclength parametrized minimizing geodesic from to . Consider the unique point in the image of satisfying . Write for the level set of constant from and suppose that . Hessian comparison implies that there exists a constant so that if , then . Put
i.e. the -ball about in . From the triangle inequality we see that
In either case, volume comparison applied to the ball establishes the desired result. ∎
It remains to show that the boundary harmonic radius is bounded from below.
For any there exists , depending only upon , so that for any , the boundary harmonic radius .
We proceed by contradiction. If the conclusion were false, then there exists a sequence so that . Choose points satisfying and consider the normalized sequence
Note that and so Theorems 2.1 and 2.2 imply that the sequence converges weakly in to a limit . Write and for the metrics on induced respectively from and . Due to the scaling properties of the various quantities we see that
The metrics have bounded curvature, diameter and volume so that passing to a subsequence if necessary we can assume (as in the proof of Lemma 2.2) that converges to in and . The limit satisfies (in the sense)
As noted in , when expressed in boundary harmonic coordinates on one obtains the system of equations (writing for the moment)
for and . Here is a polynomial in and its first derivatives.
We remark again that these equations must be (initially)
interpreted in the sense even though we did not write them in this way.
Let us show that is a smooth Riemannian manifold with smooth metric tensor (in fact it turns out that . As remarked above, so that [3, Proposition 5.2.2] applied to equations (3.14)-(3.15) shows that for some . Similarly we can obtain regularity for the other metric components by applying [3, Proposition 5.4.1] to the equations (3.11)-(3.13). In  it is shown that, in harmonic coordinates, the system
is an elliptic boundary value problem. Here is the pointwise conformal class of . Since , and (and since ),
elliptic regularity (see for instance [18, Theorem 6.8.3]) shows that in a neighborhood of .
The corresponding interior estimates are well known (see
for instance ).
Now we can show that the convergence is in the strong topology. Fix a term and set . Supress the subscript on for the moment, setting . We first look at the tangential components , with the dirichlet boundary condition . We work in boundary harmonic coordinates centered at the origin, and assume that the coordinates contain a half-ball of a definite (Euclidean) size that maps to a region in of a definite size, independent of . Let be another half-ball. The estimates below are valid on and depend upon the distance from to . From  we have the estimate
Here is a fixed positive number that can be chosen independent of . depends on , , and the norm of and therefore can be chosen independent of . The norm is the standard norm on the dual space of . See  for the definition of the norm on the trace space . The strong convergence implies in particular that
The weak convergence shows that
It remains to estimate the term
We have that weakly in (recall that in harmonic coordinates) so that
We have that in and