Curvature estimates for stable free boundary minimal hypersurfaces

Curvature estimates for stable free boundary minimal hypersurfaces

ByQiang Guangat Santa Barbara,Martin Man-chun Liat Hong Kong and Xin Zhouat Santa Barbara

Abstract. In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the cel- ebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curva- ture 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 Riemann- ian manifolds for any dimension and codimension. For3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori 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.

1. Introduction

Let.M^m; g/be anm-dimensional Riemannian manifold andNⁿM^man embedded n-dimensional submanifold called the constraint submanifold. If we consider the k-dimen- sional area functional on the space of immersedk-submanifolds†^k M^mwith boundary𝜕† lying on the constraint submanifoldN, the critical points are calledfree boundary minimal submanifolds. These are minimal submanifolds †M meeting N orthogonally along 𝜕†

(cf. Definition 2.2). Such a critical point is said to bestable(cf. Definition 2.4) if it minimizes area up to second order. The purpose of this paper is three-fold. First, we prove uniform curva- ture estimates (Theorem 1.1) forimmersedstable free boundary minimal hypersurfaces satis- fying a uniform area bound. Second, we prove a monotonicity formula (Theorem 3.4) near the

The work of Martin Man-chun Li described in this paper is substantially supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 24305115) and partially supported by direct grants from CUHK (Project Code 4053118 and 3132705). Xin Zhou is partially supported by NSF grant DMS-1406337.

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 set- ting) to establish a stronger curvature estimate (Theorem 1.2) forproperly embedded stable free boundary minimal surfaces in compact Riemannian3-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 [18]. Such curvature estimates have profound applications in the theory of minimal hypersurfaces. For example, Pitts [14] made use of Schoen–Simon–Yau’s estimates in an essential way to establish the regularity of minimal hypersurfaces†constructed by min-max methods, for2dim†5 due to the dimension restriction in [18]. Shortly after, Schoen and Simon [17] generalized these curvature estimates to any dimension (but still for codimension one, i.e.hypersurfaces) forembedded stable minimal hypersurfaces, which enabled them to complete Pitts’ program for dim† > 5.

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 [18]. We refer the readers to Section 2 for the precise definitions.

Theorem 1.1. Assume2n6. LetM^nC1be a Riemannian manifold andNⁿM an embedded hypersurface, both without boundary. Suppose thatU M is an open subset with compact closure. If .†;𝜕†/#.U; N \U / is an immersed (embedded when nD6) stable free boundary minimal hypersurface withArea.†/C0, then

jA^†j².x/ C1

dist_M² .x;𝜕U / for allx 2†;

whereC1 > 0is a constant depending only onC0,U andN\U.

An important consequence of Theorem 1.1 is a smooth compactness theorem for stable free boundary minimal hypersurfaces which arealmost properly embedded(cf. [13]). As in the work [14], 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 [13] by the last two authors. We remark that any compact Riemannian manifold  with boundary𝜕DN can be extended to a closed Riemannian manifoldM withas 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 esti- mates do not hold, we can apply a blow-up argument to a sequence of counterexamples together with a reflection principle to obtain a nonflat complete stable immersed minimal hypersurface

†₁ in R^nC1 without boundary. We then apply the Bernstein theorem in [18, Theorem 2]

(which only holds for2n5) or [17, Theorem 3] (whennD6for embedded hypersurface) to conclude that†₁is flat, hence resulting in a contradiction. Using Ros’ estimates [15, Theo- rem 9 and Corollary 11] for one-sided stable minimal surfaces, our result also holds true when nD2if one removes the two-sided condition. Whenn7, the stable free boundary minimal hypersurface may contain a singular set with Hausdorff codimension at least seven. This fol- lows from similar arguments as in [17]. 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 N. By an isometric embedding ofM into some Euclidean spaceR^L, we establish a monotonicity formula (Theo- rem 3.4) for free boundary minimal submanifolds relative toEuclideanballs ofR^Lcentered at points on the constraint submanifoldN.

We remark that Grüter and Jost proved in [10, Theorem 3.1] a version of monotonic- ity formula and used it to establish an important Allard-type regularity theorem for varifolds with free boundary. However, the monotonicity formula they obtained contains an extra term involving the mass of the varifold in a reflected ball (whose center may not lie on the constraint submanifoldN), which makes it difficult to apply in some situations (in [13] 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 [4] and [11]). We would like to mention that other monotonicity formulas have been proved for free boundary minimal submanifolds in a Euclidean unit ball (see [3, 21]).

Consider now the case of a compact Riemannian3-manifoldM with boundary𝜕M; by the remark in the paragraph after Theorem 1.1, we can assume thatM is a compact subdomain of a larger Riemannian manifoldfM without boundary andN D𝜕M is the constraint submani- fold. Furthermore, if we assume that the free boundary minimal surface†isproperly embedded inM (i.e.†M and†\𝜕M D𝜕†), then we prove a stronger uniform curvature estimate similar to the one in Theorem 1.1, butindependent of the area of†.

Theorem 1.2. Let.M³; g/be a compact Riemannian3-manifold with nonempty bound- ary𝜕M. Then there exists a constantC2 > 0depending only on the geometry ofM and𝜕M such that if.†;𝜕†/.M;𝜕M /is a compact, properly embedded stable minimal surface with free boundary, then

sup

x2†jAj².x/C2:

Remark 1.3. For simplicity, we assume that†is compact in Theorem 1.2. This ensures that†has no boundary points lying in the interior ofM. Without the compactness assump- tion, similar uniform estimates still hold as long as we stay away from the points in†n† inside the interior ofM 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 stableimmersed minimal surfaces in3-manifolds by Schoen [16] (see also [6] and [15]), we conjecture that the embeddedness of†is unnecessary.

Conjecture 1.4. 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 dimen- sion 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 Riemannian3-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 depending only on the second fundamental form of the minimal surfaces.

2. Free boundary minimal submanifolds

In this section, we give the definition of free boundary minimal submanifolds (Defini- tion 2.2) and the notion ofstability(Definition 2.4) in the hypersurface case. We also prove a reflection principle (Lemma 2.6) which will be useful in subsequent sections.

Let.M; g/be anm-dimensional Riemannian manifold, and letN M be an embedded n-dimensional constraint submanifold. We will always assume thatM; N are smooth without boundary unless otherwise stated. Suppose that†is a k-dimensional smooth manifold with boundary𝜕†(possibly empty).

Definition 2.1. We use.†;𝜕†/#.M; N /to denote an immersion' W†!M such that'.𝜕†/N. If, furthermore,'is an embedding, we denote it as.†;𝜕†/.M; N /. An embedded submanifold.†;𝜕†/.M; N /is said to beproperif'.†/\N D'.𝜕†/.

Definition 2.2. We say that.†;𝜕†/.M; N / is an immersed (resp. embedded)free boundary minimal submanifoldif

(i) ' W†!M is a minimal immersion (resp. embedding), (ii) †meetsN orthogonally along𝜕†.

Remark 2.3. Condition (ii), is often called thefree boundary condition. Note that both conditions (i) and (ii) are local properties.

Free boundary minimal submanifolds can be characterized variationally as critical points to thek-dimensional area functional of.M; g/among the class of all immersedk-submanifolds .†;𝜕†/#.M; N /. Given a smooth one-parameter family of immersions

't W.†;𝜕†/#.M; N /; t2. ; /;

whose variation vector field

X.x/D d dt

ˇ ˇ ˇ ˇ_t

D0

't.x/

is compactly supported in†, thefirst variational formula(cf. [6, Section 1.3]) says that d

dt ˇ ˇ ˇ ˇ_t

D0

Area.'t.†//D Z

†

div†X da D Z

†

XH daC Z

𝜕†

X ds;

(2.1)

whereH is the mean curvature vector of the immersion'0W†!M with outward unit conor- mal,daanddsare the induced measures on†and𝜕†, respectively. Since't.𝜕†/N for allt, it follows that the variation vector fieldX must be tangent toN along 𝜕†. Therefore, 'W.†;𝜕†/#.M; N /is a free boundary minimal submanifold if and only if (2.1) vanishes for all compactly supported variational vector fieldXwithX.p/2TpNfor allp 2𝜕†, 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 bestableif the second variation is nonnegative. For simplicity and our purpose, we will only consider thehypersurfacecase, i.e. dim†DdimN Dm 1, wheremDdimM. Recall that an immersion' W†!M is said to betwo-sidedif there exists a globally defined continuous unit normal vector fieldon†.

Definition 2.4. Let 'W.†;𝜕†/#.M; N / be an immersed free boundary minimal hypersurface. Then' is said to bestableif it is two-sided and satisfies the stability inequality, i.e.

0 d² dt²

ˇ ˇ ˇ ˇ_t

D0

Area.'t.†//

D Z

†jr^†fj² .jA^†j²CRic.; //f²da Z

𝜕†

A^N.; /f²ds;

where't W.†;𝜕†/#.M; N /is any compactly supported variation of'0D'with variation fieldX Df ,A^†andA^N are the second fundamental forms of†andN inM, respectively, and Ric is the Ricci curvature ofM.

Remark 2.5. The sign convention ofA^N in Definition 2.4 is taken such thatA^N 0 ifN D𝜕is the boundary of a convex domain inM.

One particularly important example is as follows:M DR^nC1andN DRⁿD ¹x1D0º. LetRⁿ_C^C1D ¹x10ºand let WR^nC1!R^nC1be the reflection map acrossRⁿ. We have the following reflection principle that relates free boundary minimal hypersurfaces with minimal hypersurfaces without boundary.

Lemma 2.6 (Reflection principle). If .†;𝜕†/#.R^nC1;Rⁿ/ is an immersed stable free boundary minimal hypersurface, then†[.†/ is an immersed stable minimal hyper- surface (without boundary) inR^nC1.

Proof. Since minimality is preserved under the isometry of R^nC1 and that † is orthogonal toRⁿ along𝜕†, it follows that†[.†/is aC¹minimal hypersurface inR^nC1 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 the stability inequality of Definition 2.4 vanishes forN DRⁿ.

3. Monotonicity formula

In this section, we prove a monotonicity formula (Theorem 3.4) for stationary vari- folds with free boundary (cf. 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 always assume thatM is isometrically embedded into the Euclidean spaceR^L(such an embedding exists by Nash’s isometric embedding theorem) andN M is a closedn-dimensional constraint submanifold. We will denoteB.p; r/e to be the open Euclidean ball inR^Lwith centerpand radiusr > 0.¹⁾The second fundamental form ofM inR^Lis denoted byA^M.

We begin with a discussion on the notion ofstationary varifolds with free boundary. Let Vk.M /denote the closure (with respect to the weak topology) of rectifiablek-varifoldsinR^L which is supported inM (cf. [14, 2.1(18)(g)]). As usual, theweightof a varifoldV 2V_k.M / is denoted bykVk. We refer the readers to the standard reference [19] on varifolds.

We use X.M; N / to denote the space of smooth vector fields X compactly supported onR^Lsuch thatX.x/2TxM for allx2M andX.p/2TpN for allp 2N. Any such vector fieldX 2X.M; N /generates a one-parameter family of diffeomorphismst WM !M with t.N /DN and the first variation of a varifoldV 2V_k.M /alongX is defined by

ıV .X /WD d dt

ˇ ˇ ˇ ˇ_t

D0

k.t/_]Vk.M /;

where.t/_]V 2V_k.M /denotes the pushforward of the varifoldV by the diffeomorphismt

(cf. [14, 2.1(18)(h)]).

Definition 3.1. A k-varifoldV 2V_k.M /is said to be stationary with free boundary onN ifıV .X /D0for allX 2X.M; N /.

This generalizes the notion of free boundary minimal submanifolds to allow singulari- ties. By the first variation formula for varifolds [19, Section 39.2], ak-varifoldV 2Vk.M /is stationary with free boundary onN if and only if

(3.1)

Z

G_k.R^L/

divSX.x/ d V .x; S /D0

for all X 2X.M; N /. IfX is not tangent to M butX.p/2TpN for all p2N, then (3.1) implies that

(3.2)

Z

Gk.R^L/

divSX.x/ d V .x; S /D Z

Gk.R^L/

X.x/trSA^Md V .x; S /;

whereS TxM is an arbitraryk-plane, and trSA^M D

k

X

iD1

A^M.ei; ei/

for an orthonormal basis¹e1; : : : ; e_kºofS.

1) Note that our notation is different from that in [10] whereeBis used to denote some kind of reflected ball.

The key idea to derive our monotonicity formula near a base point p2N is to find a special test vector fieldX which is asymptotic (near p) to the radial vector field centered atpand, at the same time, tangential along the constraint submanifoldN. Our choice ofX is largely motivated by [2, 10]. For convenience of the readers, we provide details on some of the preliminary results below.

Let us review some local geometry of thek-dimensional closed constraint submanifold N inR^L essentially following the discussions in [2, Section 2]. We always identify a linear subspaceP R^Lwith its orthogonal projectionP 2Hom.R^L;R^L/onto this subspace. Using this notion, we define the maps; WN !Hom.R^L;R^L/to be

.p/WDTpN and .p/WD.TpN /^?;

where TpN is the tangent space of N in R^L, and .TpN /^? is the orthogonal complement ofTpN inR^L.

To bound the turning ofN insideR^L, we define as in [2] a global geometric quantity WDinf

²

t 0W j.x/.y x/j t

2jy xj²for allx; y 2N

³ :

By the compactness and smoothness ofN, 2Œ0;1/and thus one can define the radius of curvature forN to be

(3.3) R0WD ¹ 2.0;1:

Letbe the nearest point projection map ontoNand let./WDdist_R^L.; N /be the distance function toNinR^L, both defined on a tubular neighborhood ofN. More precisely, if we define the open set

AWD [

p2N

B.p; Re 0/

which is an open neighborhood ofN insideR^L, we have the following from [2, Lemma 2.2].

Lemma 3.2. With the definitions as above,,,,are well-defined and smooth onA.

Moreover, we have the following estimates:

kDp.v/k jvj for allp2N; v 2TpN;

(3.4)

kDak 1

1 .a/ for alla2A;

(3.5)

j .a/ pj ja pj

1 ja pj for allp2N; a2eB.p; R0/:

(3.6)

Proof. See [2, Lemma 2.2].

From now on, we fix a pointp 2N. Without loss of generality, we can assume thatp D0 after a translation inR^L. By Lemma 3.2, we can define a smooth map WeB.0; R0/!R^Lby

.x/WD . .x// .x/:

Note that .x/ is the normal component (with respect to T.x/N) of the vector .x/ p (which is equal to .x/whenpD0). See Figure 1.

Figure 1. Definition of.

Lemma 3.3. Fix anys2.0; R0/. If we let D _2.R^R₀⁰_s/2, then

(3.7) kDxk 2jxj and j.x/j jxj² for allx2eB.0; s/:

Proof. Fixs2.0; R0/and anyx2eB.0; s/. AsDx.v/2T_.x/N for anyv2R^L, we have. .x//Dx.v/D0for anyv, thus

Dx.v/D ŒD_.x/ıDx.v/. .x// . .x//Dx.v/

D ŒD_.x/ıDx.v/. .x//:

Therefore, we have by (3.4), (3.5), (3.6),.x/ jxjandDx.v/2T_.x/N, kDxk 1

1 .x/ jxj

1 jxj R0

.R0 jxj/²jxj 2jxj:

The estimate forj.x/jfollows from a line integration fromx D0using that.0/D0.

We can now state our monotonicity formula.

Theorem 3.4(Monotonicity formula). Assume thatM is an embeddedm-dimensional submanifold in R^L with second fundamental form A^M bounded by some constant ƒ > 0, i.e.jA^Mj ƒ. Suppose that N M is a closed embedded n-dimensional submanifold, and V 2V_k.M / is a stationary k-varifold with free boundary on N. For any pointp2N and 0 < < < ¹₂R0as defined in(3.3), we have

e^ƒ1kVk.eB.p; //

^k e^ƒ1kVk.eB.p; //

^k

Z

Gk.e^A.p;;//

e^ƒ1rjr_S^?rj²

.1C r/r^k d V .x; S /:

Here D _R²₀ is defined in Lemma3.3 (withsD ¹₂R0),ƒ1WDk.ƒC3 /,r.x/WD jx pj, r_S^?r is the projection ofrr to the orthogonal complementS^? of thek-planeS R^L, and G_k.eA.p; ; //WDeA.p; ; /G.L; k/is the restriction of thek-dimensional Grassmannian onR^Lrestricted toeA.p; ; /WDB.p; /e neB.p; /.

Proof. As before, we can assume thatp D0by a translation inR^L. The monotonicity formula will be obtained by choosing a suitable test vector fieldX in (3.2). Define

X.x/WD'.r/.xC.x//;

wherer D jxjand' 0is a smooth cutoff function with'⁰0, and'.r/D0forr ¹₂R0. Whenx2N, we have .x/Dxand thus

xC.x/Dx .x/xD .x/x2TxN:

HenceX.x/2TxN for allx2N, and (3.2) holds true for suchX.

For anyk-dimensional subspaceS R^L, by the definition ofX, divSX.x/D'.r/.div_SxCdivS.x//C'⁰.r/r^Sr.xC.x//

D'.r/.kCdivS.x//C'⁰.r/Œr.1 jrS^?rj²/C r^Sr.x/:

By (3.7), we have the estimates

jdivS.x/j kkDxk 2k r and jr^Sr.x/j j.x/j r²: Using the fact that' 0and'⁰0, we have the following estimates:

divSX.x/'.r/.k 2k r/C'⁰.r/Œr.1 jrS^?rj²/C r² and

jX.x/j '.r/.jxj C j.x/j/'.r/.rC r²/:

Plugging these estimates into (3.2) and using the boundjA^Mj ƒ, Z

'⁰.r/r.1C r/ dkVk Ck Z

'.r/ dkVk

Z

'⁰.r/rjrS^?rj²d V .x; S /Ckƒ Z

†

'.r/r.1C r/ dkVk C2k Z

†

'.r/r dkVk: Fix a smooth cutoff function WŒ0;1/!Œ0; 1 such that ⁰0 and .s/D0 for s1.

For any2.0;¹₂R0/, if we define '.r/D.^r/, then it is a cutoff function satisfying all the assumptions above. Moreover,r'⁰.r/D _d^d'.^r/. Plugging into the inequality above, using the fact that.^r/D0forr,

.1C/d d

Z

r

Ck

Z

r

d d

Z

r

jrS^?rj²Ckƒ.1C/ Z

r

C2k Z

r

:

AddingkR

.^r/to both sides of the inequality, we obtain .1C/d

d Z

r

Ck.1C/ Z

r

d d

Z

r

jrS^?rj²CkŒƒ.1C/C3  Z

r

:

DenoteI./DR

.^r/dkVkandJ./DR

.^r/jr_S^?rj²d V .x; S /. Then we have .1C/d

d I./

^k

J⁰./

^k kŒƒ.1C/C3 I./

^k ; which clearly implies

d d

I./

^k

Ck.ƒC3 /I./

^k J⁰./

.1C/^k: Therefore, we can rewrite it into the form

d d

e^{k.ƒC3 /}I./

^k

e^{k.ƒC3 /} .1C/^kJ⁰./:

The monotonicity formula follows by letting approach the characteristic function of the intervalŒ0; 1.

4. Curvature estimates

In this section, we prove our main curvature estimates (Theorem 4.1) which imply Theo- rem 1.1. The estimates hold for immersed stable free boundary minimal hypersurfaces in any closed Riemannian manifold.M; g/with constraint hypersurfaceN M. Moreover, the esti- mates are local anduniformin the sense that the constants only depend on the geometry ofM andN, and the area of the minimal hypersurface. As in the previous section, we will continue to assume that the.nC1/-dimensional closed Riemannian manifold.M^nC1; g/is isometrically embedded intoR^LandN M is a compact embedded hypersurface inM with𝜕N D ;.

Denote B.p; r/M as the open geodesic ball ofM centered at p with radius r > 0.

Since the intrinsic distance onM and the extrinsic distance onR^Lare equivalent near a given pointp2M, we can without loss of generality assume that the monotonicity formula (Theo- rem 3.4) holds true for geodesic balls when the radius is less than someR0> 0 (depending only on.M; N / and the embedding toR^L). Now we can state our main curvature estimates near the boundary.

Theorem 4.1. Let 2n6. Suppose that M^nC1R^L, N and R0 are given as above. Letp 2N and0 < R < R0. If.†;𝜕†/#.B.p; R/; N \B.p; R// is an immersed (embedded whennD6) stable free boundary minimal hypersurface satisfying the area bound Area.†\B.p; R//C0, then

sup

x2†\B.p;^R₂/

jA^†j.x/C1;

whereC1 > 0is a constant depending onC0,M andN.

Proof. 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 blow-up 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 blow-up 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 that the assertion is false; then there exists a sequence

.†i;𝜕†i/#.B.p; R/; N \B.p; R//

of immersed (embedded whennD6) stable free boundary minimal hypersurfaces such that

(4.1) Area.†i \B.p; R//C0;

but asi ! 1, we have

sup

x2†i\B.p;^R₂/

jA^†ij.x/! 1:

Therefore, we can pick a sequence of points xi 2†i \B.p;^R₂/ such that jA^†ij.xi/! 1. By compactness we can assume thatxi !x 2B.p;^2R₃ /. By the Schoen–Simon–Yau interior curvature estimates [18] (or Schoen–Simon’s curvature estimates [17] whennD6), we must havex 2N, and moreover, the connected component of†i \B.p; R/that passes throughxi

must have a nonempty free boundary component lying onN \B.p; R/. Define a sequence of

Figure 2. The ballsB.xi; ri/andB.yi; r_i⁰/.

positive numbers

ri WD.jA^†ij.xi// ¹²:

Then we haveri !0andrijA^†ij.xi/! 1asi ! 1. Now, chooseyi 2†i\B.xi; ri/so that it achieves the maximum of

(4.2) sup

y2†i\B.xi;ri/

jA^†ij.y/distM.y;𝜕B.xi; ri//:

Letr_i⁰WDri distM.yi; xi/(see Figure 2). Note thatr_i⁰!0 asr_i⁰ri !0. Moreover, the same pointyi 2†i \B.xi; ri/also achieves the maximum of

(4.3) sup

y2†i\B.yi;r_i⁰/

jA^†ij.y/distM.y;𝜕B.yi; r_i⁰//:

Definei WD jA^†ij.yi/. Then we havei ! 1sincer_i⁰!0and ir_i⁰D jA^†ij.yi/distM.yi;𝜕B.yi; r_i⁰//

D jA^†ij.yi/distM.yi;𝜕B.xi; ri//

jA^†ij.xi/distM.xi;𝜕B.xi; ri//

DrijA^†ij.xi/! C1; where the inequality above follows from (4.2).

Let i WR^L!R^L be the blow-up maps i.z/WDi.z yi/ centered at yi. Denote .M_i⁰; N_i⁰/WD.i.M /; i.N //and letB⁰.0; r/be the open geodesic ball inM_i⁰of radiusr > 0 centered at02M_i⁰. We get a blow-up sequence of immersed stable free boundary minimal hypersurfaces

.†⁰_i;𝜕†⁰_i/WD.i.†i/; i.𝜕†i//#.B⁰.0; iR/; N⁰\B⁰.0; iR//:

Note that we havejA^†0ij.0/D_i ¹jA^†ij.yi/D1 for everyi, and the connected component of†⁰_i passing through0 must have nonempty free boundary lying onN_i⁰\B⁰.0; iR/. For each fixedr > 0, we have_i ¹r < r_i⁰for alli sufficiently large sinceir_i⁰! C1. Hence, if x2†⁰_i \B⁰.0; r/, then_i¹.x/2†i \B.yi; _i ¹r/†i\B.yi; r_i⁰/. Using (4.3), we have

(4.4) jA^†0ij.x/ ir_i⁰

ir_i⁰ r

since distM._i ¹.x/;𝜕B.yi; r_i⁰//r_i⁰ _i ¹rfor allisufficiently large (depending on the fixed r > 0). Note that the right hand side of (4.4) approaches1asi ! 1.

Step 2: The contradiction argument. By the smoothness ofM and thatyi !x 2M, we clearly haveB⁰.0; ir_i⁰/converging toTxM smoothly and locally uniformly inR^L. How- ever, as yi does not necessarily lie on N, we have to consider two cases of convergence scenario:

Case I: lim infi!1idist_R^L.yi; N /D 1.

Case II: lim infi!1idist_R^L.yi; N / <1.

For Case I, the rescaled constraint surfaceN⁰\B⁰.0; iR/will escape to infinity asi ! 1and therefore disappear in the limit. For Case II, after passing to a subsequence,N⁰\B⁰.0; iR/

smoothly and locally uniformly converge to somen-dimensional affine subspaceP R^L. Assume for now that the blow-ups †⁰_i satisfy a uniform Euclidean area growth with respect to the geodesic balls in Mi, i.e. there exists a uniform constantC2> 0such that for each fixedr > 0, wheni is sufficiently large (depending possibly onr), we have

(4.5) Area.†⁰_i \B⁰.0; r//C2rⁿ:

By using either the classical convergence theorem for minimal submanifolds with bounded curvature (for Case I) or Theorem 6.1 (for Case II), there exists a subsequence of the connected component of†⁰_i passing through0converging smoothly and locally uniformly to either

a complete, immersed stable minimal hypersurface†¹₁ inTxM, or

an immersed stable free boundary minimal hypersurface.†²₁;𝜕†²₁/#.TxM; P /such that𝜕†²₁¤ ;,

satisfying the same Euclidean area growth as in (4.5)for allr > 0with†⁰_i replaced by†¹₁ or†²₁. WhennD6, †¹₁; †²₁ are both embedded by our assumption. In the first case, the classical Bernstein theorem [18, Theorem 2] (when 2n5) or [17, Theorem 3] (when nD6) implies that†¹₁is a flat hyperplane inTxM, which is a contradiction asjA^†11j.0/D1.

In the second case, as the constraint hypersurfaceP is a hyperplane inTxM, we can double

†²₁as in Lemma 2.6 by reflecting acrossP to obtain a complete, immersed (embedded when nD6) stable minimal hypersurface inTxM 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 †⁰_i in (4.5). This is essentially a consequence of the monotonicity formula (Theorem 3.4). In the following,C3; C4; : : :will be used to denote constants depending only on.M; N /and the embeddingM R^L.

Letdi WDdistM.yi; N /and letzi 2N be the nearest point projection (inM) ofyi toN. Hencedi !0by the choice ofyi. We have to consider two cases:

Case 1: lim infi!1idi D 1.

Case 2: lim infi!1idi <1.

Let us first consider Case 1. Fixr > 0. Sinceidi ! 1, we have for alli sufficiently large (depending onr),

(4.6) B.yi; _i ¹r/B.yi; di/B.zi; 2di/B.zi;^R₂/B.p; R/:

Note thatB.yi; di/\N D ;, by the interior monotonicity formula [19, Theorem 17.6] and the

inclusions in (4.6), we have fori sufficiently large, Area.†i \B.yi; _i ¹r//C3

Area.†i\B.yi; di//

d_iⁿ ._i ¹r/ⁿ:

Using di !0, (4.6) and the boundary monotonicity formula (Theorem 3.4), we have for i sufficiently large,

Area.†i \B.yi; _i ¹r//2ⁿC4

Area.†i \B.zi;^R₂//

.^R₂/ⁿ ._i ¹r/ⁿ:

Finally, using (4.6) and (4.1), fori sufficiently large we have

Area.†i \B.yi; _i¹r//.2²ⁿC4C0R ⁿ/._i¹r/ⁿ; which implies (4.5). This finishes the proof for Case 1.

Now we consider Case 2, i.e.idi is uniformly bounded for alli. By a similar argument as above, we have

B.yi; _i ¹r/B.zi; di C_i ¹r/B.zi;^R₂/B.p; R/

for alli sufficiently large (for any fixedr > 0). By exactly the same arguments as in Case 1, we have

Area.†i \B.yi; _i¹r//C02ⁿC5R ⁿ

1C_id_i r

n

._i¹r/ⁿ:

Since idi is uniformly bounded, forr sufficiently large independent of i, estimate (4.5) is satisfied. This proves Case 2 and thus completes the proof of Theorem 4.1.

5. Proof of Theorem 1.2

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 sur- faces, we may not get a single stable minimal surface in the blow-up limit. Nonetheless, with the extraembeddedness assumption, the blow-up sequence would still subsequentially con- verge to aminimal lamination. Roughly speaking, a minimal lamination in a3-manifoldM³ is a disjoint collectionLof embedded minimal surfaces ƒ(called theleaves of the lamina- tion) such thatS

ƒ2Lƒ is a closed subset ofM. In [5], Colding and Minicozzi proved that a sequence of minimal laminations with uniformly bounded curvature subsequentially con- verges 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 M³ to be a compact 3-manifold with bound- ary𝜕M, and without loss of generality, suppose that M is a compact subdomain of another closed Riemannian3-manifoldMf. We denote the half-space

R³_CWD ¹.x¹; x²; x³/2R³Wx¹0º;

whose boundary is given by the planeR²₁D𝜕R³_CD ¹x¹D0º. First, let us recall the definition of minimal lamination from [5].

Definition 5.1 ([5, Appendix B]). Let Mf be an open subset. A minimal lami- nation of  is a collection L of disjoint, embedded, connected minimal surfaces, denoted byƒ (called theleavesof the lamination) such that[ƒ2Lƒ is a closed subset of. More- over, for eachx2, there exists a neighborhoodU ofxinand a local chart.U; ˆ/with ˆ.U /R³ so that in these coordinates the leaves in L pass throughˆ.U /in slices of the form.R² ¹tº/\ˆ.U /.

Now we can define minimal laminations with free boundary.

Definition 5.2. Aminimal lamination ofM³with free boundary on𝜕M is a collection L of disjoint, embedded, connected minimal surfaces with (possibly empty) free boundary on𝜕M, denoted byƒ, such thatS

ƒ2Lƒis a closed subset ofM. Moreover, for eachx 2M, one of the following holds:

(i) x2M n𝜕M and there exists an open neighborhoodU ofxinM n𝜕M such that the set

¹ƒ\U Wƒ2Lºis a minimal lamination ofU.

(ii) x2𝜕Mand there exist a relatively open neighborhoodUeofxinMand a local coordinate chart.eU ;eˆ/such thateˆ.eU /R³_Candˆ.e 𝜕M \U /e 𝜕R³_Cso that in these coordinates the leaves inLpass through the chart in slices of the form.R² ¹tº/\eˆ.eU /.

(iii) x2𝜕M and there exists an open neighborhoodU ofxinfM such that¹ƒ\U Wƒ2Lº is a minimal lamination ofU.

Remark 5.3. Note that the leavesƒof the laminationLin Definition 5.2 may not be properly embedded inM. For example,ƒmay touch𝜕M in the interior ofƒin case (iii).

In the special case that M³DR³_C, by the maximum principle [6, Corollary 1.28] we know that all leaves of the lamination L are properly embedded (except when ƒD𝜕R³_C).

Therefore Lemma 2.6 implies the following reflection principle for minimal lamination with free boundary.

Lemma 5.4(Lamination reflection principle). IfLis a minimal lamination ofR³_Cwith free boundary on𝜕R³_C, then¹ƒ[.ƒ/Wƒ2Lºis a minimal lamination ofR³(in the sense of Definition5.1).

We need the following convergence result, whose proof is postponed until Section 7.

Theorem 5.5. Let .M³; g/ be a compact Riemannian 3-manifold with nonempty boundary𝜕M. IfLi is a sequence of minimal laminations ofM with free boundary on𝜕M with uniformly bounded curvature, i.e. there exists a constantC > 0such that

sup¹jA^ƒj².x/Wx2ƒ2Liº C;

then a subsequence ofLi converges in theC^˛topology for any˛ < 1to a Lipschitz lamina- tionLwith minimal leaves inM and free boundary on𝜕M.

Proof of Theorem1.2. We follow the same contradiction argument as in the proof of Theorem 4.1 and adopt the same notions therein. After a blow-up process, we again face two

types of convergence scenario. By Colding–Minicozzi’s convergence theorem for minimal lam- inations with bounded curvature [5, Proposition B.1] (for Case I) and Theorem 5.5 (for Case II), a subsequence of blow-ups converges to

a minimal laminationLQ inTxM 'R³, or

a minimal laminationLin a half-spaceH 'R³_Cwith free boundary on𝜕H.

In the second case, we can apply the lamination reflection principle (Lemma 5.4) to obtain a minimal laminationLQ inTxM 'R³. By the blow-up assumption, we know that the origin 02R³is in the support ofL, and the curvature of the leafQ ƒ0passing through0is exactly1 at0, i.e.jA^ƒ0j.0/D1.

Now we analyze the structure of the minimal laminationLQ R³for both cases. We refer to [12] for well-known terminologies for minimal laminations. Ifƒ2 QL is an accumulating leaf, then eitherƒor its double coverƒQ is a complete, stable minimal surface inR³, which must be an affine plane by the Bernstein theorem inR³(see [7, 8]). Therefore, the leafƒ0passing through0must be an isolated leaf. Since all the surfaces in the sequence†⁰_i are stable with free boundary, the smooth convergence of†⁰_i toLQ orLand the reflection principle (Lemma 2.6) imply thatƒ₀is a complete, stable, minimal surface inR³. This again violates the Bernstein theorem asjA^ƒ0j.0/D1by our construction. Therefore, we arrive at a contradiction and finish the proof of Theorem 1.2.

6. Convergence of free boundary minimal submanifolds

In this section, we prove a general convergence result (Theorem 6.1) for free boundary minimal submanifolds with uniformly bounded second fundamental form. Note that this con- vergence result does not require stability and holds in any dimension and codimension.

To facilitate our discussion, let us first review some basic properties of Fermi coordi- nates. LetNⁿM^nC1 be an embedded hypersurface (without boundary) in the Riemannian manifold.M; g/. We can assume that bothN andM are complete. Fix a pointp2N. We let .x1; : : : ; xn/be the geodesic normal coordinates ofN centered atp, and lett DdistM.; N / be the signed distance function fromN which is well-defined and smooth in a neighborhood ofp insideM. Therefore, forr0> 0sufficiently small, there exists a diffeomorphism, called aFermi coordinate chart,

WB_rⁿ₀^C1.0/TpM !U M;

.t; x1; : : : ; xn/7!.t; x1; : : : ; xn/;

such thatU \N D.¹t D0º/. Here,B_rⁿ₀^C1.0/is the open Euclidean ball ofTpM ŠR^nC1 of radiusr0> 0centered at0. We refer the readers to [13] for a more detailed discussion on Fermi coordinates. The components of the metricgin Fermi coordinates satisfygt t D1and gx_it D0fori D1; : : : ; n.

Let.†;𝜕†/.M; N /be smooth embedded free boundary minimalk-dimensional sub- manifold, with1kn. Fix anyp2𝜕†N, and let WB_rⁿ₀^C1.0/!U be a Fermi coordi- nate chart as above centered atp. After a rotation we can assume that

Tp.𝜕†/D ¹x_k D DxnD0Dtº ŠR^{k 1}:

Since†meetsN orthogonally along𝜕†, after picking a choice on the sign oft, the tangent half-spaceTp†is given by

Tp†D ¹x_k D DxnD0; t 0º ŠR^k_C:

Hence, under the Fermi coordinates in a neighborhood ofp,†can be written as a graph of uD.u1; : : : ; unC1 k/which is anR^{nC1 k}-valued function of.t; x⁰/D.t; x1; : : : ; xk 1/in a domain ofR^k_C, i.e. ¹.†/D ¹.t; x⁰; u.t; x⁰//º Rⁿ_C^C1. Moreover, ¹.𝜕†/is given by the same graph witht D0. Since _𝜕t^𝜕 is a unit normal vector field alongN\U, it is clear that the free boundary condition along𝜕†is equivalent to

(6.1) 𝜕u_`

𝜕t .0; x⁰/D0 for`D1; : : : ; nC1 k.

We now state the convergence result for free boundary minimal submanifolds with uni- formly bounded area and the second fundamental form.

Theorem 6.1. Suppose we have a sequence .†j;𝜕†j/#.M^nC1; Nⁿ/ of immersed free boundary minimalk-dimensional submanifolds, where1kn, with uniformly bounded area and second fundamental form, i.e. there exist positive constantsC0; C1 > 0such that

Area.†j/C0 and sup

†j

jA^†jj C1

for all j, then after passing to a subsequence, .†j;𝜕†j/ converges smoothly and locally uniformly to .†₁;𝜕†₁/#.M; N / which is a smooth immersed free boundary minimal k-dimensional submanifold.

Proof. The convergence away from N follows from the classical convergence results.

By the second fundamental form bound, we can coverN by balls (of a uniform size) under Fermi coordinates centered atp2N such that each†j can be written as graphs over some domain ofTp†j with uniformly bounded gradient (see [6, Section 2.2]). By using the uniform area bound together with the monotonicity formula (Theorem 3.4), there is a uniform upper bound on the number of sheets of the graphs. After passing to a subsequence, the number of sheets remains constant for all j and each sheet is a graph over a k-dimensional subspace ofTpM or ak-dimensional half-space orthogonal toTpN. The first case again follows from the classical interior convergence result. The second case follows from standard elliptic PDE theory with Neumann boundary conditions (6.1) (see [1] for example).

7. Convergence of free boundary minimal lamination

Finally, we give the proof of Theorem 5.5 which was used in Section 5.

Proof of Theorem5.5. For simplicity we will assume that each lamination Li has finitely many leaves where the number of leaves may depend on i; this will suffice for our application. For any interior pointx2M n𝜕M, the argument used in the proof of [5, Propo- sition B.1] implies the convergence in a small neighborhood ofxinM n𝜕M. Hence, we only need to deal with the convergence near a boundary pointx2𝜕M.

Fix p2𝜕M and let N D𝜕M. The theorem will follow once we construct uniform coordinate charts in a small neighborhood ofpin the Fermi coordinate system as in Section 6.

Let' be a Fermi coordinate chart in a relatively open neighborhoodU ofpinM, i.e.

' WU M !Ue R³_C;

such that'.p/D0 and '.N \U /D ¹x1D0º \Ue. Here, .x1; x2; x3/ are the local Fermi coordinate system centered atp (i.e. tDx1). Suppose that B_4r^C

0 Ue for some small r0 to be chosen later, whereB_4r^C

0 DB4r₀\ ¹x1 0ºdenotes the half ball inR³_Cwith radius 4r0

centered at the origin.

Next, we will construct uniform coordinate charts on' ¹.B_r^C₀/. Note that for eachiand everyƒ2Li, we have sup_ƒjA^ƒj² C:We may chooser0 sufficiently small so thatC r0is as small as we wish. Then for each fixedi,

[

ƒ2L_i

'.ƒ\U /\B_4r^C

0

gives a finite number of disconnected surfaces with bounded curvature in the Fermi coordinate system.

Since the lamination has uniformly bounded curvature, by the tilt estimates [6, Lem- ma 2.4], there exists a constantı > 0such that for each laminationLi, we have the following two cases:

(i) None of the leaves ofLi meets𝜕R³_CinB_{ı r}^C

0 (except possibly for one leaf touching𝜕R³_C tangentially at some points).

(ii) There exists a leaf ofLi meeting𝜕R³_Calong some nonempty free boundary.

For case (i), we can construct uniform coordinate charts as in the proof of [5, Proposition B.1]

in a neighborhood of the larger manifoldfM. For case (ii), we claim that inB_{2ı r}^C

0, all leaves ofLi which intersect B_{ı r}^C

0 must meet𝜕R³_Calong some nonempty free boundary; otherwise, the tilt estimates will imply that two leaves intersect somewhere inB_r^C₀ which contradicts the assumption that all leaves are disjoint. Note that the tilt estimates in [6, Lemma 2.4] only use the uniform curvature bound of leaves inLi, but not the minimal surface equations.

Now, we focus on case (ii). For simplicity, we user0 to denoteır0. The free boundary condition and the choice of Fermi coordinates imply that these surfaces meet𝜕R³_Corthogonally in the Euclidean metric. Going to a further subsequence (possibly withr0 even smaller), for fixedi, every sheet of

[

ƒ2Li

'.ƒ\U /\B_2r^C

0

which intersectsB_r^C₀is a graph with small gradient over a subset of certain fixed plane perpen- dicular to𝜕R³_C(which can be chosen asR² ¹0º WD ¹x³D0ºafter a rotation keeping𝜕R³_C fixed as a set) containing a half ball of radiusr0(see [6, Lemma 2.4]).

We will show that in a concentric half ball of smaller radius inB_2r^C

0, the sequence of lam- inations converges in theC^˛topology to a lamination for any˛ < 1. The coordinate chartˆ required by the definition of a lamination will be given by the Arzela–Ascoli theorem as a limit of a sequence of bi-Lipschitz maps

ˆi WB_2r^C₀ !R³_C

with bounded bi-Lipschitz constants, and ˆwill be defined on a slightly smaller concentric half ballB_{s r}^C₀for somes > 0to be determined. Furthermore, we will show that for eachifixed

ˆi

B_{s r}^C₀\'

[

ƒ2Li

ƒ\U

is the union of subsets of planes which are each parallel toR² ¹0º R³_C. Set the mapˆi by letting

ˆ_i¹.y1; y2; y3/D.y1; y2; i.y1; y2; y3//;

wherei is defined as follows: order the sheets ofB_2r^C

0\'.S

ƒ2Liƒ\U /asƒ_i;kfork1by increasing values ofx3and letƒ_i;kbe the graph of the functionf_i;kover (part of) theR² ¹0º plane. In the following we only need to consider those sheets ƒi;k where ƒi;k\B_r^C₀ ¤ ;, since we eventually will work on a much smaller concentric half ball. The domain of suchf_i;k contains the half ball of radiusr0centered at the origin of the R² ¹0ºplane. Again asC r0

can be chosen small enough, we can assume thatjrf_i;kjare as small as we want. Moreover, the free boundary condition satisfied byƒi;kis equivalent to the Neumann boundary condition

𝜕fi;k.0;/

𝜕x1 D0:

Set w_i;k Df_i;kC1 f_i;k. In the following, , r, and div will be with respect to the Euclidean metric onR² ¹0º. By a standard computation (cf. [6, Chapter 7] or [20, equa- tion (7)]), we have

(7.1) div..aCId/rw_i;k/Cbrw_i;kCcw_i;k D0 and 𝜕wi;k.0;/

𝜕x1 D0;

whereais a matrix-valued function,bis a vector-valued function, andcis simply a real-valued function.

Note thata,b, andc depend oni, but the norms ofa; b; ccan be made uniformly small ifC r0is small enough and if we rescale our ambient manifold by a large factor. By (7.1), and the Harnack inequality (see [9, Section 8.20] and [1, Section 6]) applied to the positive function wi;kgives

(7.2) sup

B_{2s r0}^C

w_i;kC1 inf

B^C_{2s r0}

w_i;k;

whereC1depends only on the norms ofa; bandc. Here,B^C_t is the half ball inR² ¹0ºwith radiustand center 0. SetMi;kDf_i;k.0; 0/. In the region¹.y1; y2; y3/2B^C_r0ŒM_i;k;Mi;kC1º, define the functioni by

i.y1; y2; y3/Df_i;k.y1; y2/C y3 Mi;k

M_i;kC1 M_i;kw_i;k.y1; y2/:

Hence,

ˆ_i¹.y1; y2; f_i;k.0; 0//D.y1; y2; f_i;k.y1; y2//I that is,ˆi mapsƒi;kto a subset of the planeR² ¹fi;k.0; 0/º.