Curvature estimates for stable free boundary minimal hypersurfaces
In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalizes the celebrated Schoen-Simon-Yau interior curvature estimates  up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors . We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For -manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-prioi area bound. This generalizes Schoen’s interior curvature estimates  to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi in .
Let be an -dimensional Riemannian manifold, and be an embedded -dimensional submanifold called the constraint submanifold. If we consider the -dimensional area functional on the space of immersed -submanifolds with boundary lying on the constraint submanifold , the critical points are called free boundary minimal submanifolds. These are minimal submanifolds meeting orthogonally along (c.f. Definition 2.2). Such a critical point is said to be stable (c.f. Definition 2.4) if it minimizes area up to second order. The purpose of this paper is three-fold. First, we prove uniform curvature estimates (Theorem 1.1) for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound. Second, we prove a monotonicity formula (Theorem 3.4) near the boundary for free boundary minimal submanifolds in any dimension and codimension. Finally, we use Colding-Minicozzi’s theory of minimal laminations (adapted to the free boundary setting) to establish a stronger curvature estimate (Theorem 1.2) for properly embedded stable free boundary minimal surfaces in compact Riemannian -manifolds with boundary, without assuming a uniform area bound on the minimal surfaces.
Curvature estimates for immersed stable minimal hypersurfaces in Riemannian manifolds were first proved in the celebrated work of Schoen, Simon and Yau in . Such curvature estimates have profound applications in the theory of minimal hypersurfaces. For example, Pitts  made use of Schoen-Simon-Yau’s estimates in an essential way to establish the regularity of minimal hypersurfaces constructed by min-max methods, for due to the dimension restriction in . Shortly after, Schoen and Simon  generalized these curvature estimates to any dimension (but still for codimension one, i.e. hypersurfaces) for embedded stable minimal hypersurfaces, which enabled them to complete Pitts’ program for .
In this paper, we establish uniform curvature estimates in the free boundary setting. The theorem below follows from our curvature estimates near the free boundary (Theorem 4.1) and the interior curvature estimates .
Assume . Let be a Riemannian manifold and be an embedded hypersurface. Suppose is an open subset. If is an immersed (embedded when ) stable (two-sided) free boundary minimal hypersurface with , then
where is a constant depending only on , and .
An important consequence of Theorem 1.1 is a smooth compactness theorem for stable free boundary minimal hypersurfaces which are almost properly embedded (c.f. [13, Theorem 2.15]). As in , this is a key ingredient in the regularity part of the min-max theory for free boundary minimal hypersurfaces in compact Riemannian manifolds with boundary, which is developed in  by the last two authors. We remark that any compact Riemannian manifold with boundary can be extended to a closed Riemannian manifold with as a compact domain. Hence, our curvature estimates above can be applied in this situation as well.
Our proof of the curvature estimates uses a contradiction argument. If the curvature estimates do not hold, we can apply a blow-up argument to a sequence of counterexamples together with a reflection principle to obtain a non-flat complete stable immersed minimal hypersurface in without boundary. We then apply the Bernstein Theorem in [16, Theorem 2] (which only holds for ) or [18, Theorem 3] (when for embedded hypersurface) to conclude that is flat, hence resulting in a contradiction. Using Ros’s estimates [15, Theorem 9 and Corollary 11] for one-sided stable minimal surfaces, our result also holds true when if one removes the two-sided condition. When , the stable free boundary minimal hypersurface may contain a singular set with Hausdorff codimension at least seven. This follows from similar arguments as in . To keep this paper less technical, the details will appear in a forthcoming paper.
The classical monotonicity formula plays an important role in the regularity theory for minimal submanifolds, even without the stability assumption. Unfortunately, it ceases to hold once the ball hits the boundary of the minimal submanifold. Therefore, to study the boundary regularity of free boundary minimal submanifolds, we need a monotonicity formula which holds for balls centered at points lying on the constraint submanifold . By an isometric embedding of into some Euclidean space , we establish a monotonicity formula (Theorem 3.4) for free boundary minimal submanifolds relative to Euclidean balls of centered at points on the constraint submanifold .
We remark that Grüter and Jost proved in  a version of monotonicity formula (Theorem 3.1 in ) and used it to establish an important Allard-type regularity theorem for varifolds with free boundary. However, the monotonicity formula they obtained [10, Theorem 3.1] contains an extra term involving the mass of the varifold in a reflected ball, which makes it difficult to apply in some situations (in  for example). In contrast, our monotonicity formula (Theorem 3.4) does not require any reflection which makes it more readily applicable. Moreover, the formula holds in the Riemannian manifold setting for stationary varifolds with free boundary in any dimension and codimension. We expect that our monotonicity formula might be useful in the regularity theory for other natural free boundary problem in calibrated geometries (see for example  and ). We would like to mention that other monotonicity formulas have been proved for free boundary minimal submanifolds in a Euclidean unit ball (, ).
Consider now the case of a compact Riemannian -manifold with boundary , by the remark in the paragraph after Theorem 1.1, we can assume that is a compact subdomain of a larger Riemannian manifold without boundary and is the constraint submanifold. Furthermore, if we assume that the free boundary minimal surface is properly embedded in (i.e. and ), then we prove a stronger uniform curvature estimate similar to the one in Theorem 1.1, but independent of the area of .
Let be a compact Riemannian 3-manifold with boundary . Then there exists a constant depending only on the geometry of and , such that if is a compact, properly embedded stable minimal surface with free boundary, then
For simplicity, we assume that is compact in Theorem 1.2. This ensures that has no boundary points lying in the interior of . Without the compactness assumption, similar uniform estimates still hold as long as we stay away from the points in inside the interior of as in Theorem 1.1. Note that is always locally two-sided under the embeddedness assumption.
Our proof of Theorem 1.2 involves the theory of minimal laminations which require the minimal surface to be embedded. In view of the celebrated interior curvature estimates for stable immersed minimal surfaces in -manifolds by Schoen  (see also  and ), we conjecture that the embeddedness of is unnecessary.
Theorem 1.2 holds even when is immersed.
The organization of the paper is as follows. In section 2, we give the basic definitions for free boundary minimal submanifolds in any dimension and codimension and discuss the notion of stability in the hypersurface case. In section 3, we prove the monotonicity formula (Theorem 3.4) for stationary varifolds with free boundary near the free boundary in any dimension and codimension. In section 4, we prove our main curvature estimates (Theorem 4.1) for stable free boundary minimal hypersurfaces near the free boundary. In section 5, we prove the stronger curvature estimate (Theorem 1.2) in the case of properly embedded stable free boundary minimal surfaces in a Riemannian -manifold with boundary. In section 6, we prove a general convergence result for free boundary minimal submanifolds (in any dimension and codimension) satisfying uniform bounds on area and the second fundamental form. Finally, in section 7, we prove a lamination convergence result for free boundary minimal surfaces in a three-manifold with uniform bound only on the second fundamental form of the minimal surfaces.
Acknowledgements: The authors would like to thank Prof. Richard Schoen for his continuous encouragement. They also want to thank Prof. Shing Tung Yau, Prof. Tobias Colding and Prof. Bill Minicozzi for their interest in this work. M. Li is partially supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No.: CUHK 24305115] and CUHK Direct Grant [Project Code: 4053118]. X. Zhou is partially supported by NSF grant DMS-1406337. The authors are grateful for the anonymous referee for valuable comments.
2. Free Boundary Minimal Submanifolds
In this section, we give the definition of free boundary minimal submanifolds (Definition 2.2) and the notion of stability (Definition 2.4) in the hypersurface case. We also prove a reflection principle (Lemma 2.6) which will be useful in subsequent sections.
Let be an -dimensional Riemannian manifold, and be an embedded -dimensional constraint submanifold. We will always assume are smooth without boundary unless otherwise stated. Suppose is a -dimensional smooth manifold with boundary (possibly empty).
We use to denote an immersion such that . If, furthermore, is an embedding, we denote it as . An embedded submanifold is said to be proper if .
We say that is an immersed (resp. embedded) free boundary minimal submanifold if
is a minimal immersion (resp. embedding), and
meets orthogonally along .
Condition (ii), is often called the free boundary condition. Note that both conditions (i) and (ii) are local properties.
Free boundary minimal submanifolds can be characterized variationally as critical points to the -dimensional area functional of among the class of all immersed -submanifolds . Given a smooth -parameter family of immersions , , whose variation vector field is compactly supported in , the first variational formula (c.f. [6, §1.13]) says that
where is the mean curvature vector of the immersion with outward unit conormal , and are the induced measures on and respectively. Since for all , the variation vector field must be tangent to along . Therefore, is a free boundary minimal submanifold if and only if (2.1) vanishes for all compactly supported variational vector field with for all , which is equivalent to conditions (i) and (ii) in Definition 2.2.
Since free boundary minimal submanifolds are critical points to the area functional, we can look at the second variation and study their stability. Roughly speaking, a free boundary minimal submanifold is said to be stable if the second variation is non-negative. For simplicity and our purpose, we will only consider the hypersurface case, i.e. . Recall that an immersion is said to be two-sided if there exists a globally defined continuous unit normal vector field on .
An immersed free boundary minimal hypersurface is said to be stable if it is two-sided and satisfies the stability inequality, i.e.
where is any compactly supported variation of with variation field , and are the second fundamental forms of and in respectively, and is the Ricci curvature of .
The sign convention of in (2.2) is taken such that if is the boundary of a convex domain in .
One particularly important example is and . Let and be the reflection map across . We have the following reflection principle that relates free boundary minimal hypersurfaces with minimal hypersurfaces without boundary.
Lemma 2.6 (Reflection principle).
If is an immersed stable free boundary minimal hypersurface, then is an immersed stable minimal hypersurface (without boundary) in .
Since minimality is preserved under the isometry of and that is orthogonal to along , is a minimal hypersurface in without boundary. Higher regularity for minimal hypersurfaces implies that it is indeed smooth across . Stability follows directly from the definition since the boundary term in (2.2) vanishes for . ∎
3. Monotonicity formula
In this section, we prove a monotonicity formula (Theorem 3.4) for stationary varifolds with free boundary (c.f. Definition 3.1) in Riemannian manifolds for any dimension and codimension. The monotonicity formula for free boundary minimal submanifolds is then a direct corollary.
Throughout this section, we will consider as an embedded -dimensional submanifold (by Nash isometric embedding theorem) and a compact closed -dimensional constraint submanifold . We will denote to be the open Euclidean ball in with center and radius . The second fundamental form of in is denoted by .
We begin with a discussion on the notion of stationary varifolds with free boundary. Let denote the closure (with respect to the weak topology) of rectifiable -varifolds in which is supported in (c.f. [14, 2.1(18)(g)]). As usual, the weight of a varifold is denoted by . We refer the readers to the standard reference  on varifolds.
We use to denote the space of smooth vector fields compactly supported on such that for all and for all . Any such vector field generates a one-parameter family of diffeomorphisms with and the first variation of a varifold along is defined by
where is the pushforward of by the diffeomorphism (c.f. [14, 2.1(18)(h)]).
A -varifold is said to be stationary with free boundary on if for all .
This generalizes the notion of free boundary minimal submanifolds to allow singularities. By the first variation formula for varifolds [19, 39.2], a -varifold is stationary with free boundary on if and only
for all . If is not tangent to but for all , then (3.1) implies that
where is an arbitrary -plane, and for an orthonormal basis of .
The key idea to derive our monotonicity formula near a base point is to find a special test vector field which is asymptotic (near ) to the radial vector field centered at and, at the same time, tangential along the constraint submanifold . Our choice of is largely motivated by [2, 10], and we add the following preliminary results for completeness.
Let us review some local geometry of the -dimensional compact closed constraint submanifold in essentially following the discussions in [2, §2]. We always identify a linear subspace with its orthogonal projection onto this subspace. Using this notion, we define the maps to be
where is the tangent space of in , and is the orthogonal complement of in .
To bound the turning of inside , we define as in  a global geometric quantity
By the compactness and smoothness of , and thus one can define the radius of curvature for to be
Let be the nearest point projection map onto and be the distance function to in , both defined on a tubular neighborhood of . More precisely, if we define the open set
which is an open neighborhood of inside , we have the following lemma from [2, Lemma 2.2].
With the definitions as above, , , , are well-defined and smooth on . Moreover, we have the following estimates:
See [2, Lemma 2.2]. ∎
From now on, we fix a point . Without loss of generality, we can assume that after a translation in . By Lemma 3.2, we can define a smooth map by
Note that is the normal component (with respect to ) of the vector (which is equal to when ). See Figure 1.
Fix any , if we let , then
We can now state our monotonicity formula.
Theorem 3.4 (Monotonicity Formula).
Let be an embedded -dimensional submanifold in with second fundamental form bounded by some constant , i.e. . Suppose is a compact, closed, embedded -dimensional submanifold, and is a stationary -varifold with free boundary on .
As before, we can assume by a translation in . The monotonicity formula will be obtained by choosing a suitable test vector field in (3.2). Define
where and is a smooth cutoff function with , and for . When , we have and thus
Hence for all , and (3.2) holds true for such .
For any -dimensional subspace , by the definition of ,
By (3.8), we have the estimates
Using the fact that and , we have the following estimates
Plugging these estimates into (3.2) and using the bound ,
Fix a smooth cutoff function such that and for . For any , if we define , then it is a cutoff function satisfying all the assumptions above. Moreover, . Plugging into the inequality above, using the fact that for ,
Adding to both sides of the inequality, we obtain
Denote and , then we have
which clearly implies
Therefore, we can rewrite it into the form
The monotonicity formula follows by letting approach the characteristic function of . ∎
4. Curvature estimates
In this section, we prove our main curvature estimates (Theorem 4.1) which imply Theorem 1.1. The estimates hold for immersed stable free boundary minimal hypersurfaces in any closed Riemannian manifold with constraint hypersurface . Moreover, the estimates are local and uniform in the sense that the constants only depend on the geometry of and , and the area of the minimal hypersurface. Throughout this section, we will assume that the -dimensional closed Riemannian manifold is isometrically embedded into and is a compact embedded hypersurface in with .
Denote as the open geodesic ball of centered at with radius . Since the intrinsic distance on and the extrinsic distance on are equivalent near a given point , we can WLOG assume that the monotonicity formula (Theorem 3.4) holds true for geodesic balls when the radius is less than some (depending only on and the embedding to ). Now we can state our main curvature estimates near the boundary.
Let . Suppose are given as above. Let and . If is an immersed (embedded when ) stable free boundary minimal hypersurface satisfying the area bound: , then
where is a constant depending on , and .
The proof is by a contradiction argument which will be divided into three steps. First, if the assertion is false, then we can carry out a blowup argument to obtain a limit after a suitable rescaling. Second, we show that if the limit satisfies certain area growth condition, it has to be a flat hyperplane which would give a contradiction to the choice of the blowup sequence. Finally, we check that the limit indeed satisfies the area growth condition using the monotonicity formula (Theorem 3.4).
Step 1: The blow-up argument.
Suppose the assertion is false, then there exists a sequence of immersed (embedded when ) stable free boundary minimal hypersurfaces such that
but as , we have
Therefore, we can pick a sequence of points such that . By compactness we can assume that . By Schoen-Simon-Yau interior curvature estimates  (or Schoen-Simon’s curvature estimates  when ), we must have , and moreover, the connected component of that passes through must have a non-empty free boundary component lying on . Define a sequence of positive numbers
then we have and as . Now, choose so that it achieves the maximum of
Let . Note that as . (See Figure 2) Moreover, the same point also achieves the maximum of
Define , then we have since and
where the inequality above follows from (4.2).
Let be the blow up maps centered at . Denote and be the open geodesic ball in of radius centered at . We get a blow-up sequence of immersed stable free boundary minimal hypersurfaces
Note that we have for every , and the connected component of passing through must have non-empty free boundary lying on . For each fixed , we have for all sufficiently large since . Hence, if , then . Using (4.3), we have
since for all sufficiently large (depending on the fixed ). Note that the right hand side of (4.4) approaches as .
Step 2: The contradiction argument.
By the smoothness of and that , we clearly have converging to smoothly and locally uniformly in . However, as does not necessarily lie on , we have to consider two types of convergence scenario:
Type I: ,
Type II: .
For Type I convergence, the rescaled constraint surface will escape to infinity as and therefore disappear in the limit. For Type II convergence, after passing to a subsequence, smoothly and locally uniformly to some -dimensional affine subspace .
Assume for now that the blow-ups satisfy a uniform Euclidean area growth with respect to the geodesic balls in , i.e., there exists a uniform constant such that for each fixed , when is sufficiently large (depending possibly on ), we have
Using either the classical convergence theorem for minimal submanifolds with bounded curvature (for Type I convergence) or Theorem 6.1 (for Type II convergence), there exists a subsequence of the connected component of passing through converging smoothly and locally uniformly to either
a complete, immersed stable minimal hypersurface in , or
a non-compact, immersed stable free boundary minimal hypersurface such that ,
satisfying the same Euclidean area growth as in (4.5) for all with replaced by or . When , are both embedded by our assumption. In the first case, the classical Bernstein Theorem [16, Theorem 2] (when ) or [18, Theorem 3] (when ) implies that is a flat hyperplane in , which is a contradiction as . In the second case, as the constraint hypersurface is a hyperplane in , we can double as in Lemma 2.6 by reflecting across to obtain a complete, immersed (embedded when ) stable minimal hypersurface in with Euclidean area growth. This gives the same contradiction as in the first case.
Step 3: The area growth condition.
It remains now to establish the uniform Euclidean area growth for in (4.5). This is essentially a consequence of the monotonicity formula (Theorem 3.4). In the following, will be used to denote constants depending only on .
Let and be the nearest point projection (in ) of to . Hence by the choice of . We have to consider two cases:
Case 1: ,
Case 2: .
Let us first consider Case 1. Fix . Since , we have for all sufficiently large (depending on )
which implies (4.5). This finishes the proof for Case 1.
Now we consider Case 2, i.e. is uniformly bounded for all . By similar argument as above, we have
for all sufficiently large (for any fixed ). By exactly the same arguments as in Case 1, we have
In this section, we prove Theorem 1.2 using the same blow-up arguments as in the proof of Theorem 4.1. However, since we do not assume a uniform area bound of the minimal surfaces, we may not get a single stable minimal surface in the blow-up limit. Nonetheless, with the extra embeddedness assumption, the blow-up sequence would still subsequentially converge to a minimal lamination. Roughly speaking, a minimal lamination in a 3-manifold is a disjoint collection of embedded minimal surfaces (called the leaves of the lamination) such that is a closed subset of . In , Colding and Minicozzi proved that a sequence of minimal laminations with uniformly bounded curvature subsequentially converges to a limit minimal lamination. For our purpose, we will generalize the notion of minimal laminations to include the case with free boundary.
Throughout this section, we will denote to be a compact -manifold with boundary , and without loss of generality, suppose that is a compact subdomain of another closed Riemannian -manifold . Moreover, we denote the half-space