Curvature estimates for stable free boundary minimal hypersurfaces
CCURVATURE ESTIMATES FOR STABLE FREEBOUNDARY MINIMAL HYPERSURFACES
QIANG GUANG, MARTIN MAN-CHUN LI, AND XIN ZHOU
Abstract.
In this paper, we prove uniform curvature estimates for im-mersed stable free boundary minimal hypersurfaces satisfying a uniformarea bound, which generalizes the celebrated Schoen-Simon-Yau inte-rior curvature estimates [16] up to the free boundary. Our curvatureestimates imply a smooth compactness theorem which is an essential in-gredient in the min-max theory of free boundary minimal hypersurfacesdeveloped by the last two authors [13]. We also prove a monotonicity for-mula for free boundary minimal submanifolds in Riemannian manifoldsfor any dimension and codimension. For 3-manifolds with boundary, weprove a stronger curvature estimate for properly embedded stable freeboundary minimal surfaces without a-prioi area bound. This generalizesSchoen’s interior curvature estimates [17] to the free boundary setting.Our proof uses the theory of minimal laminations developed by Coldingand Minicozzi in [5]. Introduction
Let (
M, g ) be an m -dimensional Riemannian manifold, and N be an em-bedded n -dimensional submanifold called the constraint submanifold . Ifwe consider the k -dimensional area functional on the space of immersed k -submanifolds Σ ⊂ M with boundary ∂ Σ lying on the constraint subman-ifold N , the critical points are called free boundary minimal submanifolds .These are minimal submanifolds Σ ⊂ M meeting N orthogonally along ∂ Σ(c.f. Definition 2.2). Such a critical point is said to be stable (c.f. Definition2.4) if it minimizes area up to second order. The purpose of this paper isthree-fold. First, we prove uniform curvature estimates (Theorem 1.1) for immersed stable free boundary minimal hypersurfaces satisfying a uniformarea bound. Second, we prove a monotonicity formula (Theorem 3.4) nearthe boundary for free boundary minimal submanifolds in any dimensionand codimension. Finally, we use Colding-Minicozzi’s theory of minimallaminations (adapted to the free boundary setting) to establish a strongercurvature estimate (Theorem 1.2) for properly embedded stable free bound-ary minimal surfaces in compact Riemannian 3-manifolds with boundary, without assuming a uniform area bound on the minimal surfaces.Curvature estimates for immersed stable minimal hypersurfaces in Rie-mannian manifolds were first proved in the celebrated work of Schoen, Si-mon and Yau in [16]. Such curvature estimates have profound applications a r X i v : . [ m a t h . DG ] A ug QIANG GUANG, MARTIN LI, AND XIN ZHOU in the theory of minimal hypersurfaces. For example, Pitts [14] made useof Schoen-Simon-Yau’s estimates in an essential way to establish the reg-ularity of minimal hypersurfaces Σ constructed by min-max methods, for2 ≤ dim Σ ≤ hypersurfaces ) for embedded stableminimal hypersurfaces, which enabled them to complete Pitts’ program fordim Σ > Theorem 1.1.
Assume ≤ n ≤ . Let M n +1 be a Riemannian manifoldand N n ⊂ M be an embedded hypersurface. Suppose U ⊂⊂ M is an opensubset. If (Σ , ∂ Σ) ⊂ ( U, N ∩ U ) is an immersed (embedded when n = 6 )stable (two-sided) free boundary minimal hypersurface with Area(Σ) ≤ C ,then | A Σ | ( x ) ≤ C dist M ( x, ∂U ) for all x ∈ Σ , where C > is a constant depending only on C , U and N ∩ U . An important consequence of Theorem 1.1 is a smooth compactness theo-rem for stable free boundary minimal hypersurfaces which are almost prop-erly embedded (c.f. [13, Theorem 2.15]). As in [14], this is a key ingredientin the regularity part of the min-max theory for free boundary minimalhypersurfaces in compact Riemannian manifolds with boundary, which isdeveloped in [13] by the last two authors. We remark that any compactRiemannian manifold Ω with boundary ∂ Ω = N can be extended to a closedRiemannian manifold M with Ω as a compact domain. Hence, our curvatureestimates above can be applied in this situation as well.Our proof of the curvature estimates uses a contradiction argument. Ifthe curvature estimates do not hold, we can apply a blow-up argument toa sequence of counterexamples together with a reflection principle to ob-tain a non-flat complete stable immersed minimal hypersurface Σ ∞ in R n +1 without boundary . We then apply the Bernstein Theorem in [16, Theorem2] (which only holds for 2 ≤ n ≤
5) or [18, Theorem 3] (when n = 6 forembedded hypersurface) to conclude that Σ ∞ is flat, hence resulting in acontradiction. Using Ros’s estimates [15, Theorem 9 and Corollary 11] forone-sided stable minimal surfaces, our result also holds true when n = 2 ifone removes the two-sided condition. When n ≥
7, the stable free boundaryminimal hypersurface may contain a singular set with Hausdorff codimen-sion at least seven. This follows from similar arguments as in [18]. To keepthis paper less technical, the details will appear in a forthcoming paper.The classical monotonicity formula plays an important role in the regu-larity theory for minimal submanifolds, even without the stability assump-tion. Unfortunately, it ceases to hold once the ball hits the boundary of
URVATURE ESTIMATES NEAR FREE BOUNDARY 3 the minimal submanifold. Therefore, to study the boundary regularity offree boundary minimal submanifolds, we need a monotonicity formula whichholds for balls centered at points lying on the constraint submanifold N . Byan isometric embedding of M into some Euclidean space R L , we establisha monotonicity formula (Theorem 3.4) for free boundary minimal submani-folds relative to Euclidean balls of R L centered at points on the constraintsubmanifold N .We remark that Gr¨uter and Jost proved in [10] a version of monotonic-ity formula (Theorem 3.1 in [10]) and used it to establish an importantAllard-type regularity theorem for varifolds with free boundary. However,the monotonicity formula they obtained [10, Theorem 3.1] contains an ex-tra term involving the mass of the varifold in a reflected ball, which makesit difficult to apply in some situations (in [13] for example). In contrast,our monotonicity formula (Theorem 3.4) does not require any reflectionwhich makes it more readily applicable. Moreover, the formula holds in theRiemannian manifold setting for stationary varifolds with free boundary inany dimension and codimension. We expect that our monotonicity formulamight be useful in the regularity theory for other natural free boundaryproblem in calibrated geometries (see for example [4] and [11]). We wouldlike to mention that other monotonicity formulas have been proved for freeboundary minimal submanifolds in a Euclidean unit ball ([3], [21]).Consider now the case of a compact Riemannian 3-manifold M withboundary ∂M , by the remark in the paragraph after Theorem 1.1, we canassume that M is a compact subdomain of a larger Riemannian manifold (cid:102) M without boundary and N = ∂M is the constraint submanifold. Furthermore,if we assume that the free boundary minimal surface Σ is properly embedded in M (i.e. Σ ⊂ M and Σ ∩ ∂M = ∂ Σ), then we prove a stronger uniformcurvature estimate similar to the one in Theorem 1.1, but independent ofthe area of Σ. Theorem 1.2.
Let ( M , g ) be a compact Riemannian 3-manifold with bound-ary ∂M (cid:54) = ∅ . Then there exists a constant C > depending only on thegeometry of M and ∂M , such that if (Σ , ∂ Σ) ⊂ ( M, ∂M ) is a compact,properly embedded stable minimal surface with free boundary, then sup x ∈ Σ | A | ( x ) ≤ C . Remark . For simplicity, we assume that Σ is compact in Theorem 1.2.This ensures that Σ has no boundary points lying in the interior of M .Without the compactness assumption, similar uniform estimates still holdas long as we stay away from the points in Σ \ Σ inside the interior of M as in Theorem 1.1. Note that Σ is always locally two-sided under theembeddedness assumption.Our proof of Theorem 1.2 involves the theory of minimal laminationswhich require the minimal surface to be embedded . In view of the cele-brated interior curvature estimates for stable immersed minimal surfaces in QIANG GUANG, MARTIN LI, AND XIN ZHOU
Conjecture 1.4.
Theorem 1.2 holds even when Σ is immersed. The organization of the paper is as follows. In section 2, we give the basicdefinitions for free boundary minimal submanifolds in any dimension andcodimension and discuss the notion of stability in the hypersurface case. Insection 3, we prove the monotonicity formula (Theorem 3.4) for stationaryvarifolds with free boundary near the free boundary in any dimension andcodimension. In section 4, we prove our main curvature estimates (Theorem4.1) for stable free boundary minimal hypersurfaces near the free boundary.In section 5, we prove the stronger curvature estimate (Theorem 1.2) inthe case of properly embedded stable free boundary minimal surfaces in aRiemannian 3-manifold with boundary. In section 6, we prove a generalconvergence result for free boundary minimal submanifolds (in any dimen-sion and codimension) satisfying uniform bounds on area and the secondfundamental form. Finally, in section 7, we prove a lamination convergenceresult for free boundary minimal surfaces in a three-manifold with uniformbound only on the second fundamental form of the minimal surfaces.
Acknowledgements : The authors would like to thank Prof. RichardSchoen for his continuous encouragement. They also want to thank Prof.Shing Tung Yau, Prof. Tobias Colding and Prof. Bill Minicozzi for their in-terest in this work. M. Li is partially supported by a research grant from theResearch Grants Council of the Hong Kong Special Administrative Region,China [Project No.: CUHK 24305115] and CUHK Direct Grant [ProjectCode: 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 subman-ifolds (Definition 2.2) and the notion of stability (Definition 2.4) in thehypersurface case. We also prove a reflection principle (Lemma 2.6) whichwill be useful in subsequent sections.Let (
M, g ) be an m -dimensional Riemannian manifold, and N ⊂ M bean embedded n -dimensional constraint submanifold. We will always assume M, N are smooth without boundary unless otherwise stated. Suppose Σ isa k -dimensional smooth manifold with boundary ∂ Σ (possibly empty).
Definition 2.1.
We use (Σ , ∂ Σ) (cid:35) ( M, N ) to denote an immersion ϕ : Σ → M such that ϕ ( ∂ Σ) ⊂ N . If, furthermore, ϕ is an embedding, we denote itas (Σ , ∂ Σ) (cid:44) → ( M, N ). An embedded submanifold (Σ , ∂ Σ) ⊂ ( M, N ) is saidto be proper if ϕ (Σ) ∩ N = ϕ ( ∂ Σ).
Definition 2.2.
We say that (Σ , ∂ Σ) ⊂ ( M, N ) is an immersed (resp. em-bedded) free boundary minimal submanifold if URVATURE ESTIMATES NEAR FREE BOUNDARY 5 (i) ϕ : Σ → M is a minimal immersion (resp. embedding), and(ii) Σ meets N orthogonally along ∂ Σ. Remark . Condition (ii), is often called the free boundary condition . Notethat both conditions (i) and (ii) are local properties.Free boundary minimal submanifolds can be characterized variationallyas critical points to the k -dimensional area functional of ( M, g ) among theclass of all immersed k -submanifolds (Σ , ∂ Σ) ⊂ ( M, N ). Given a smooth1-parameter family of immersions ϕ t : (Σ , ∂ Σ) → ( M, N ), t ∈ ( − (cid:15), (cid:15) ), whosevariation vector field X ( x ) = ddt (cid:12)(cid:12) t =0 ϕ t ( x ) is compactly supported in Σ, the first variational formula (c.f. [6, § ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Area( ϕ t (Σ)) = (cid:90) Σ div Σ X da = − (cid:90) Σ X · H da + (cid:90) ∂ Σ X · η ds, where H is the mean curvature vector of the immersion ϕ : Σ → M withoutward unit conormal η , da and ds are the induced measures on Σ and ∂ Σ respectively. Since ϕ t ( ∂ Σ) ⊂ N for all t , the variation vector field X must be tangent to N along ∂ Σ. Therefore, ϕ : (Σ , ∂ Σ) (cid:35) ( M, N ) is a freeboundary minimal submanifold if and only if (2.1) vanishes for all compactlysupported variational vector field X with X ( p ) ∈ T p N for all p ∈ ∂ Σ, whichis equivalent to conditions (i) and (ii) in Definition 2.2.Since free boundary minimal submanifolds are critical points to the areafunctional, 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, wewill only consider the hypersurface case, i.e. dim Σ = dim N = dim M − ϕ : Σ → M is said to be two-sided if there existsa globally defined continuous unit normal vector field ν on Σ. Definition 2.4.
An immersed free boundary minimal hypersurface ϕ :(Σ , ∂ Σ) (cid:35) ( M, N ) is said to be stable if it is two-sided and satisfies thestability inequality, i.e.0 ≤ d dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Area( ϕ t (Σ))= (cid:90) Σ |∇ Σ f | − ( | A Σ | + Ric( ν, ν )) f da − (cid:90) ∂ Σ A N ( ν, ν ) f ds, (2.2)where ϕ t : (Σ , ∂ Σ) (cid:35) ( M, N ) is any compactly supported variation of ϕ = ϕ with variation field X = f ν , A Σ and A N are the second fundamentalforms of Σ and N in M respectively, and Ric is the Ricci curvature of M . Remark . The sign convention of A N in (2.2) is taken such that A N ≥ N = ∂ Ω is the boundary of a convex domain in M .One particularly important example is M = R n +1 and N = R n = { x =0 } . Let R n +1+ = { x ≥ } and θ : R n +1 → R n +1 be the reflection map across QIANG GUANG, MARTIN LI, AND XIN ZHOU R n . We have the following reflection principle that relates free boundaryminimal hypersurfaces with minimal hypersurfaces without boundary. Lemma 2.6 (Reflection principle) . If (Σ , ∂ Σ) (cid:35) ( R n +1 , R n ) is an immersedstable free boundary minimal hypersurface, then Σ ∪ θ (Σ) is an immersedstable minimal hypersurface (without boundary) in R n +1 .Proof. Since minimality is preserved under the isometry θ of R n +1 and thatΣ is orthogonal to R n along ∂ Σ, Σ ∪ θ (Σ) is a C minimal hypersurfacein R n +1 without boundary. Higher regularity for minimal hypersurfacesimplies that it is indeed smooth across ∂ Σ. Stability follows directly fromthe definition since the boundary term in (2.2) vanishes for N = R n . (cid:3) Monotonicity formula
In this section, we prove a monotonicity formula (Theorem 3.4) for sta-tionary varifolds with free boundary (c.f. Definition 3.1) in Riemannianmanifolds for any dimension and codimension. The monotonicity formulafor free boundary minimal submanifolds is then a direct corollary.Throughout this section, we will consider M ⊂ R L as an embedded m -dimensional submanifold (by Nash isometric embedding theorem) and acompact closed n -dimensional constraint submanifold N ⊂ M . We willdenote (cid:101) B ( p, r ) to be the open Euclidean ball in R L with center p and radius r >
0. The second fundamental form of M in R L is denoted by A M .We begin with a discussion on the notion of stationary varifolds with freeboundary . Let V k ( M ) denote the closure (with respect to the weak topology)of rectifiable k -varifolds in R L which is supported in M (c.f. [14, 2.1(18)(g)]).As usual, the weight of a varifold V ∈ V k ( M ) is denoted by (cid:107) V (cid:107) . We referthe readers to the standard reference [19] on varifolds.We use X ( M, N ) to denote the space of smooth vector fields X compactlysupported on R L such that X ( x ) ∈ T x M for all x ∈ M and X ( p ) ∈ T p N forall p ∈ N . Any such vector field X ∈ X ( M, N ) generates a one-parameterfamily of diffeomorphisms φ t : M → M with φ t ( N ) = N and the firstvariation of a varifold V ∈ V k ( M ) along X is defined by δV ( X ) := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:107) ( φ t ) (cid:93) V (cid:107) ( M ) , where ( φ t ) (cid:93) V ∈ V k ( M ) is the pushforward of V by the diffeomorphism φ t (c.f. [14, 2.1(18)(h)]). Definition 3.1. A k -varifold V ∈ V k ( M ) is said to be stationary with freeboundary on N if δV ( X ) = 0 for all X ∈ X ( M, N ).This generalizes the notion of free boundary minimal submanifolds toallow singularities. By the first variation formula for varifolds [19, 39.2], a k -varifold V ∈ V k ( M ) is stationary with free boundary on N if and only(3.1) (cid:90) div S X ( x ) dV ( x, S ) = 0 URVATURE ESTIMATES NEAR FREE BOUNDARY 7 for all X ∈ X ( M, N ). If X is not tangent to M but X ( p ) ∈ T p N for all p ∈ N , then (3.1) implies that(3.2) (cid:90) div S X ( x ) dV ( x, S ) = (cid:90) X ( x ) · tr S A M dV ( x, S ) , where S ⊂ T x M is an arbitrary k -plane, and tr S A M = (cid:80) ki =1 A M ( e i , e i ) foran orthonormal basis { e , · · · , e k } of S .The key idea to derive our monotonicity formula near a base point p ∈ N is to find a special test vector field X which is asymptotic (near p ) to theradial vector field centered at p and, at the same time, tangential along theconstraint submanifold N . Our choice of X is largely motivated by [2, 10],and we add the following preliminary results for completeness.Let us review some local geometry of the k -dimensional compact closedconstraint submanifold N in R L essentially following the discussions in [2, § P ⊂ R L with its orthogonalprojection P ∈ Hom( R L , R L ) onto this subspace. Using this notion, wedefine the maps τ, ν : N → Hom( R L , R L ) to be τ ( p ) := T p N and ν ( p ) := ( T p N ) ⊥ , where T p N is the tangent space of N in R L , and ( T p N ) ⊥ is the orthogonalcomplement of T p N in R L .To bound the turning of N inside R L , we define as in [2] a global geometricquantity κ := inf (cid:26) t ≥ | ν ( x )( y − x ) | ≤ t | y − x | for all x, y ∈ N (cid:27) . By the compactness and smoothness of N , κ ∈ [0 , ∞ ) and thus one candefine the radius of curvature for N to be(3.3) R := κ − ∈ (0 , ∞ ] . Let ξ be the nearest point projection map onto N and ρ ( · ) := dist R L ( · , N ) bethe distance function to N in R L , both defined on a tubular neighborhoodof N . More precisely, if we define the open set A := (cid:91) p ∈ N (cid:101) B ( p, R )which is an open neighborhood of N inside R L , we have the following lemmafrom [2, Lemma 2.2]. Lemma 3.2.
With the definitions as above, ξ , ρ , τ , ν are well-defined andsmooth on A . Moreover, we have the following estimates: (3.4) (cid:107) Dν p ( v ) (cid:107) ≤ κ | v | , ∀ p ∈ N, v ∈ T p N, (3.5) (cid:107) Dξ a (cid:107) ≤ − κρ ( a ) , ∀ a ∈ A, QIANG GUANG, MARTIN LI, AND XIN ZHOU (3.6) | ξ ( a ) − p | ≤ | a − p | − κ | a − p | , ∀ p ∈ N, a ∈ (cid:101) B ( p, R ) . Proof.
See [2, Lemma 2.2]. (cid:3)
From now on, we fix a point p ∈ N . Without loss of generality, we canassume that p = 0 after a translation in R L . By Lemma 3.2, we can definea smooth map ζ : (cid:101) B (0 , R ) → R L by(3.7) ζ ( x ) := − ν (cid:0) ξ ( x ) (cid:1) ξ ( x ) . Note that − ζ ( x ) is the normal component (with respect to T ξ ( x ) N ) of thevector ξ ( x ) − p (which is equal to ξ ( x ) when p = 0). See Figure 1. Figure 1.
Definition of ζ Lemma 3.3.
Fix any s ∈ (0 , R ) , if we let γ = R R − s ) , then (3.8) (cid:107) Dζ x (cid:107) ≤ γ | x | and | ζ ( x ) | ≤ γ | x | ∀ x ∈ (cid:101) B (0 , s ) . Proof.
Fix s ∈ (0 , R ) and any x ∈ (cid:101) B (0 , s ). As Dξ x ( v ) ∈ T ξ ( x ) N for any v ∈ R L , we have ν ( ξ ( x )) Dξ x ( v ) = 0 for any v , thus Dζ x ( v ) = − [ Dν ξ ( x ) ◦ Dξ x ( v )]( ξ ( x )) − ν ( ξ ( x )) Dξ x ( v )= − [ Dν ξ ( x ) ◦ Dξ x ( v )]( ξ ( x )) . Therefore, we have by (3.4), (3.5), (3.6), ρ ( x ) ≤ | x | and Dξ x ( v ) ∈ T ξ ( x ) N , (cid:107) Dζ x (cid:107) ≤ κ · − κρ ( x ) · | x | − κ | x | ≤ R ( R − | x | ) | x | ≤ γ | x | . The estimate for | ζ ( x ) | follows from a line integration from x = 0 using that ζ (0) = 0. (cid:3) We can now state our monotonicity formula.
Theorem 3.4 (Monotonicity Formula) . Let M be an embedded m -dimensionalsubmanifold in R L with second fundamental form A M bounded by some con-stant Λ > , i.e. | A M | ≤ Λ . Suppose N ⊂ M is a compact, closed, embedded n -dimensional submanifold, and V ∈ V k ( M ) is a stationary k -varifold withfree boundary on N .For any p ∈ N and < σ < ρ < R as defined in (3.3), we have e Λ σ (cid:107) V (cid:107) ( (cid:101) B ( p, σ )) σ k ≤ e Λ ρ (cid:107) V (cid:107) ( (cid:101) B ( p, ρ )) ρ k − (cid:90) G k ( (cid:101) A ( p,σ,ρ )) e Λ r |∇ ⊥ S r | (1 + γr ) r k dV ( x, S ) . URVATURE ESTIMATES NEAR FREE BOUNDARY 9
Here γ = R is defined in Lemma 3.3 (with s = R ), Λ := k (Λ + 3 γ ) , r ( x ) := | x − p | , ∇ ⊥ S r is the projection of ∇ r to the orthogonal comple-ment S ⊥ of the k -plane S ⊂ R L , and G k ( (cid:101) A ( p, σ, ρ )) := (cid:101) A ( p, σ, ρ ) × G ( L, k ) is the restriction of the k -dimensional Grassmannian on R L restricted to (cid:101) A ( p, σ, ρ ) := (cid:101) B ( p, ρ ) \ (cid:101) B ( p, σ ) .Proof. As before, we can assume p = 0 by a translation in R L . The mono-tonicity formula will be obtained by choosing a suitable test vector field X in (3.2). Define X ( x ) := ϕ ( r ) (cid:0) x + ζ ( x ) (cid:1) , where r = | x | and ϕ ≥ ϕ (cid:48) ≤
0, and ϕ ( r ) = 0 for r ≥ R . When x ∈ N , we have ξ ( x ) = x and thus x + ζ ( x ) = x − ν ( x ) x = τ ( x ) x ∈ T x N. Hence X ( x ) ∈ T x N for all x ∈ N , and (3.2) holds true for such X .For any k -dimensional subspace S ⊂ R L , by the definition of X ,div S X ( x ) = ϕ ( r ) (cid:0) div S x + div S ζ ( x ) (cid:1) + ϕ (cid:48) ( r ) ∇ S r · (cid:0) x + ζ ( x ) (cid:1) = ϕ ( r )( k + div S ζ ( x )) + ϕ (cid:48) ( r ) (cid:2) r (1 − |∇ ⊥ S r | ) + ∇ S r · ζ ( x ) (cid:3) . By (3.8), we have the estimates | div S ζ ( x ) | ≤ k (cid:107) Dζ x (cid:107) ≤ kγr, |∇ S r · ζ ( x ) | ≤ | ζ ( x ) | ≤ γr . Using the fact that ϕ ≥ ϕ (cid:48) ≤
0, we have the following estimatesdiv S X ( x ) ≥ ϕ ( r )( k − kγr ) + ϕ (cid:48) ( r ) (cid:2) r (1 − |∇ ⊥ S r | ) + γr (cid:3) , | X ( x ) | ≤ ϕ ( r )( | x | + | ζ ( x ) | ) ≤ ϕ ( r )( r + γr ) . Plugging these estimates into (3.2) and using the bound | A M | ≤ Λ, (cid:90) ϕ (cid:48) ( r ) r (1 + γr ) d (cid:107) V (cid:107) + k (cid:90) ϕ ( r ) d (cid:107) V (cid:107)≤ (cid:90) ϕ (cid:48) ( r ) r |∇ ⊥ S r | dV ( x, S ) + k Λ (cid:90) Σ ϕ ( r ) r (1 + γr ) d (cid:107) V (cid:107) + 2 kγ (cid:90) Σ ϕ ( r ) rd (cid:107) V (cid:107) . Fix a smooth cutoff function φ : [0 , ∞ ) → [0 ,
1] such that φ (cid:48) ≤ φ ( s ) = 0 for s ≥
1. For any ρ ∈ (0 , R ), if we define ϕ ( r ) = φ ( rρ ),then it is a cutoff function satisfying all the assumptions above. Moreover, rϕ (cid:48) ( r ) = − ρ ddρ ϕ ( rρ ). Plugging into the inequality above, using the fact that φ ( rρ ) = 0 for r ≥ ρ , − ρ (1 + γρ ) ddρ (cid:90) φ (cid:18) rρ (cid:19) + k (cid:90) φ (cid:18) rρ (cid:19) ≤ − ρ ddρ (cid:90) φ (cid:18) rρ (cid:19) |∇ ⊥ S r | + k Λ ρ (1 + γρ ) (cid:90) φ (cid:18) rρ (cid:19) + 2 kγρ (cid:90) φ (cid:18) rρ (cid:19) . Adding kγρ (cid:82) φ ( rρ ) to both sides of the inequality, we obtain − ρ (1 + γρ ) ddρ (cid:90) φ (cid:18) rρ (cid:19) + k (1 + γρ ) (cid:90) φ (cid:18) rρ (cid:19) ≤ − ρ ddρ (cid:90) φ (cid:18) rρ (cid:19) |∇ ⊥ S r | + kρ [Λ(1 + γρ ) + 3 γ ] (cid:90) φ (cid:18) rρ (cid:19) . Denote I ( ρ ) = (cid:82) φ ( rρ ) d (cid:107) V (cid:107) and J ( ρ ) = (cid:82) φ ( rρ ) |∇ ⊥ S r | dV ( x, S ), then we have(1 + γρ ) ddρ (cid:18) I ( ρ ) ρ k (cid:19) ≥ J (cid:48) ( ρ ) ρ k − k [Λ(1 + γρ ) + 3 γ ] I ( ρ ) ρ k , which clearly implies ddρ (cid:18) I ( ρ ) ρ k (cid:19) + k (Λ + 3 γ ) I ( ρ ) ρ k ≥ J (cid:48) ( ρ )(1 + γρ ) ρ k . Therefore, we can rewrite it into the form ddρ (cid:18) e k (Λ+3 γ ) ρ I ( ρ ) ρ k (cid:19) ≥ e k (Λ+3 γ ) ρ (1 + γρ ) ρ k J (cid:48) ( ρ ) . The monotonicity formula follows by letting φ approach the characteristicfunction of [0 , (cid:3) 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 freeboundary minimal hypersurfaces in any closed Riemannian manifold (
M, g )with constraint hypersurface N ⊂ M . Moreover, the estimates are local and uniform in the sense that the constants only depend on the geometry of M and N , and the area of the minimal hypersurface. Throughout this section,we will assume that the ( n + 1)-dimensional closed Riemannian manifold( M n +1 , g ) is isometrically embedded into R L and N ⊂ M is a compactembedded hypersurface in M with ∂N = ∅ .Denote B ( p, r ) ⊂ M as the open geodesic ball of M centered at p withradius r >
0. Since the intrinsic distance on M and the extrinsic distanceon R L are equivalent near a given point p ∈ M , we can WLOG assume thatthe monotonicity formula (Theorem 3.4) holds true for geodesic balls whenthe radius is less than some R > M, N ) and theembedding to R L ). Now we can state our main curvature estimates near theboundary. Theorem 4.1.
Let ≤ n ≤ . Suppose M n +1 ⊂ R L , N, R are given asabove. Let p ∈ N and < R < R . If (Σ , ∂ Σ) (cid:35) ( B ( p, R ) , N ∩ B ( p, R )) is animmersed (embedded when n = 6 ) stable free boundary minimal hypersurfacesatisfying the area bound: Area(Σ ∩ B ( p, R )) ≤ C , then sup x ∈ Σ ∩ B ( p, R ) | A Σ | ( x ) ≤ C , URVATURE ESTIMATES NEAR FREE BOUNDARY 11 where C > is a constant depending on C , M and N .Proof. The proof is by a contradiction argument which will be divided intothree steps. First, if the assertion is false, then we can carry out a blowupargument to obtain a limit after a suitable rescaling. Second, we showthat if the limit satisfies certain area growth condition, it has to be a flathyperplane which would give a contradiction to the choice of the blowupsequence. Finally, we check that the limit indeed satisfies the area growthcondition using the monotonicity formula (Theorem 3.4).
Step 1:
The blow-up argument.
Suppose the assertion is false, then there exists a sequence (Σ i , ∂ Σ i ) ⊂ ( B ( p, R ) , N ∩ B ( p, R )) of immersed (embedded when n = 6) stable freeboundary minimal hypersurfaces such that(4.1) Area(Σ i ∩ B ( p, R )) ≤ C , but as i → ∞ , we have sup x ∈ Σ i ∩ B ( p, R ) | A Σ i | ( x ) → ∞ . Therefore, we can pick a sequence of points x i ∈ Σ i ∩ B ( p, R ) such that | A Σ i | ( x i ) → ∞ . By compactness we can assume that x i → x ∈ B ( p, R ).By Schoen-Simon-Yau interior curvature estimates [16] (or Schoen-Simon’scurvature estimates [18] when n = 6), we must have x ∈ N , and moreover,the connected component of Σ i ∩ B ( p, R ) that passes through x i must have anon-empty free boundary component lying on N ∩ B ( p, R ). Define a sequenceof positive numbers r i := ( | A Σ i | ( x i )) − , then we have r i → r i | A Σ i | ( x i ) → ∞ as i → ∞ . Now, choose y i ∈ Σ i ∩ B ( x i , r i ) so that it achieves the maximum of(4.2) sup y ∈ Σ i ∩ B ( x i ,r i ) | A Σ i | ( y ) dist M ( y, ∂B ( x i , r i )) . Let r (cid:48) i := r i − dist M ( y i , x i ). Note that r (cid:48) i → r (cid:48) i ≤ r i →
0. (See Figure 2)Moreover, the same point y i ∈ Σ i ∩ B ( x i , r i ) also achieves the maximum of(4.3) sup y ∈ Σ i ∩ B ( y i ,r (cid:48) i ) | A Σ i | ( y ) dist M ( y, ∂B ( y i , r (cid:48) i )) . Define λ i := | A Σ i | ( y i ), then we have λ i → ∞ since r (cid:48) i → λ i r (cid:48) i = | A Σ i | ( y i ) dist M ( y i , ∂B ( y i , r (cid:48) i ))= | A Σ i | ( y i ) dist M ( y i , ∂B ( x i , r i )) ≥ | A Σ i | ( x i ) dist M ( x i , ∂B ( x i , r i ))= r i | A Σ i | ( x i ) → + ∞ , where the inequality above follows from (4.2).Let η i : R L → R L be the blow up maps η i ( z ) := λ i ( z − y i ) centered at y i .Denote ( M (cid:48) i , N (cid:48) i ) := ( η i ( M ) , η i ( N )) and B (cid:48) (0 , r ) be the open geodesic ball Figure 2. B ( x i , r i ) and B ( y i , r (cid:48) i )in M (cid:48) i of radius r > ∈ M (cid:48) i . We get a blow-up sequence ofimmersed stable free boundary minimal hypersurfaces(Σ (cid:48) i , ∂ Σ (cid:48) i ) := ( η i (Σ i ) , η i ( ∂ Σ i )) ⊂ ( B (cid:48) (0 , λ i R ) , N (cid:48) ∩ B (cid:48) (0 , λ i R )) . Note that we have | A Σ (cid:48) i | (0) = λ − i | A Σ i | ( y i ) = 1 for every i , and the connectedcomponent of Σ (cid:48) i passing through 0 must have non-empty free boundarylying on N (cid:48) i ∩ B (cid:48) (0 , λ i R ). For each fixed r >
0, we have λ − i r < r (cid:48) i forall i sufficiently large since λ i r (cid:48) i → + ∞ . Hence, if x ∈ Σ (cid:48) i ∩ B (cid:48) (0 , r ), then η − i ( x ) ∈ Σ i ∩ B ( y i , λ − i r ) ⊂ Σ i ∩ B ( y i , r (cid:48) i ). Using (4.3), we have(4.4) | A Σ (cid:48) i | ( x ) ≤ λ i r (cid:48) i λ i r (cid:48) i − r since dist M ( η − i ( x ) , ∂B ( y i , r (cid:48) i )) ≥ r (cid:48) i − λ − i r for all i sufficiently large (depend-ing on the fixed r > i → ∞ . Step 2:
The contradiction argument.
By the smoothness of M and that y i → x ∈ M , we clearly have B (cid:48) (0 , λ i r (cid:48) i )converging to T x M smoothly and locally uniformly in R L . However, as y i does not necessarily lie on N , we have to consider two types of convergencescenario: • Type I: lim inf i →∞ λ i dist R L ( y i , N ) = ∞ , • Type II: lim inf i →∞ λ i dist R L ( y i , N ) < ∞ .For Type I convergence, the rescaled constraint surface N (cid:48) ∩ B (cid:48) (0 , λ i R ) willescape to infinity as i → ∞ and therefore disappear in the limit. For Type IIconvergence, after passing to a subsequence, N (cid:48) ∩ B (cid:48) (0 , λ i R ) → P smoothlyand locally uniformly to some n -dimensional affine subspace P ⊂ R L .Assume for now that the blow-ups Σ (cid:48) i satisfy a uniform Euclidean areagrowth with respect to the geodesic balls in M i , i.e., there exists a uniformconstant C > r >
0, when i is sufficiently large(depending possibly on r ), we have(4.5) Area (cid:0) Σ (cid:48) i ∩ B (cid:48) (0 , r ) (cid:1) ≤ C r n . URVATURE ESTIMATES NEAR FREE BOUNDARY 13
Using either the classical convergence theorem for minimal submanifoldswith bounded curvature (for Type I convergence) or Theorem 6.1 (for TypeII convergence), there exists a subsequence of the connected component ofΣ (cid:48) i passing through 0 converging smoothly and locally uniformly to either • a complete, immersed stable minimal hypersurface Σ ∞ in T x M , or • a non-compact, immersed stable free boundary minimal hypersurface(Σ ∞ , ∂ Σ ∞ ) ⊂ ( T x M, P ) such that ∂ Σ ∞ (cid:54) = ∅ ,satisfying the same Euclidean area growth as in (4.5) for all r > (cid:48) i replaced by Σ ∞ or Σ ∞ . When n = 6, Σ ∞ , Σ ∞ are both embedded by ourassumption. In the first case, the classical Bernstein Theorem [16, Theorem2] (when 2 ≤ n ≤
5) or [18, Theorem 3] (when n = 6) implies that Σ ∞ isa flat hyperplane in T x M , which is a contradiction as A Σ ∞ (0) = 1. In thesecond case, as the constraint hypersurface P is a hyperplane in T x M , wecan double Σ ∞ as in Lemma 2.6 by reflecting across P to obtain a complete,immersed (embedded when n = 6) stable minimal hypersurface in T x M with Euclidean area growth. This gives the same contradiction as in thefirst case. Step 3:
The area growth condition.
It remains now to establish the uniform Euclidean area growth for Σ (cid:48) i in(4.5). This is essentially a consequence of the monotonicity formula (The-orem 3.4). In the following, C , C , · · · will be used to denote constantsdepending only on ( M ⊂ R L , N ).Let d i := dist M ( y i , N ) and z i ∈ N be the nearest point projection (in M )of y i to N . Hence d i → y i . We have to consider two cases: • Case 1: lim inf i →∞ λ i d i = ∞ , • Case 2: lim inf i →∞ λ i d i < ∞ .Let us first consider Case 1 . Fix r >
0. Since λ i d i → ∞ , we have for all i sufficiently large (depending on r )(4.6) B ( y i , λ − i r ) ⊂ B ( y i , d i ) ⊂ B ( z i , d i ) ⊂ B ( z i , R ⊂ B ( p, R ) . Note that B ( y i , d i ) ∩ N = ∅ , by the interior monotonicity formula [19, The-orem 17.6] and (4.6), we have for i sufficiently largeArea(Σ i ∩ B ( y i , λ − i r )) ≤ C Area(Σ i ∩ B ( y i , d i )) d ni ( λ − i r ) n . Using d i →
0, (4.6) and the boundary monotonicity formula (Theorem 3.4),we have for i sufficiently largeArea(Σ i ∩ B ( y i , λ − i r )) ≤ n C Area (cid:0) Σ i ∩ B ( z i , R ) (cid:1) ( R ) n ( λ − i r ) n . Finally, using (4.6) and (4.1), for i sufficiently large we haveArea(Σ i ∩ B ( y i , λ − i r )) ≤ (cid:0) n C C R − n (cid:1) · ( λ − i r ) n , which implies (4.5). This finishes the proof for Case 1 . Now we consider
Case 2 , i.e. λ i d i is uniformly bounded for all i . Bysimilar argument as above, we have B ( y i , λ − i r ) ⊂ B ( z i , d i + λ − i r ) ⊂ B ( z i , R ⊂ B ( p, R )for all i sufficiently large (for any fixed r > Case 1 , we haveArea(Σ i ∩ B ( y i , λ − i r )) ≤ C n C R − n (cid:18) λ i d i r (cid:19) n · ( λ − i r ) n . Since λ i d i is uniformly bounded, for r sufficiently large independent of i ,(4.5) is satisfied. This proves Case 2 and thus completes the proof of The-orem 4.1. (cid:3) Proof of Theorem 1.2
In this section, we prove Theorem 1.2 using the same blow-up argumentsas in the proof of Theorem 4.1. However, since we do not assume a uniformarea bound of the minimal surfaces, we may not get a single stable minimalsurface in the blow-up limit. Nonetheless, with the extra embeddedness assumption, the blow-up sequence would still subsequentially converge toa minimal lamination . Roughly speaking, a minimal lamination in a 3-manifold M is a disjoint collection L of embedded minimal surfaces Λ(called the leaves of the lamination) such that ∪ Λ ∈L Λ is a closed subsetof M . In [5], Colding and Minicozzi proved that a sequence of minimallaminations with uniformly bounded curvature subsequentially converges toa limit minimal lamination. For our purpose, we will generalize the notionof minimal laminations to include the case with free boundary.Throughout this section, we will denote M to be a compact 3-manifoldwith boundary ∂M , and without loss of generality, suppose that M is acompact subdomain of another closed Riemannian 3-manifold (cid:102) M . Moreover,we denote the half-space R := { ( x , x , x ) ∈ R : x ≥ } , whose boundary is given by the plane R = ∂ R = { x = 0 } . First, let usrecall the definition of minimal lamination from [5]. Definition 5.1 (Appendix B in [5]) . Let Ω ⊂ (cid:102) M be an open subset. A minimal lamination of Ω is a collection L of disjoint, embedded, connectedminimal surfaces, denoted by Λ (called the leaves of the lamination) suchthat ∪ Λ ∈L Λ is a closed subset of Ω. Moreover • for each x ∈ Ω, there exists a neighborhood U of x in Ω and a localchart ( U, Φ) with Φ( U ) ⊂ R so that in these coordinates the leavesin L pass through Φ( U ) in slices of the form ( R × { t } ) ∩ Φ( U ).Now we can define minimal laminations with free boundary. URVATURE ESTIMATES NEAR FREE BOUNDARY 15
Definition 5.2. A minimal lamination of M with free boundary on ∂M is a collection L of disjoint, embedded, connected minimal surfaces with(possibly empty) free boundary on ∂M , denoted by Λ, such that ∪ Λ ∈L Λ isa closed subset of M . Moreover, for each x ∈ M , one of the following holds:(i) x ∈ M \ ∂M and there exists an open neighborhood U of x in M \ ∂M such that { Λ ∩ U : Λ ∈ L} is a minimal lamination of U ;(ii) x ∈ ∂M and there exists a relatively open neighborhood (cid:101) U of x in M and a local coordinate chart ( (cid:101) U , (cid:101)
Φ) such that (cid:101) Φ( (cid:101) U ) ⊂ R and (cid:101) Φ( ∂M ∩ (cid:101) U ) ⊂ ∂ R so that in these coordinates the leaves in L passthrough the chart in slices of the form ( R × { t } ) ∩ (cid:101) Φ( (cid:101) U );(iii) x ∈ ∂M and there exists an open neighborhood U of x in (cid:102) M , suchthat { Λ ∩ U : Λ ∈ L} is a minimal lamination of U . Remark . Note that the leaves Λ of L in Definition 5.2 may not be prop-erly embedded in M . For example, Λ may touch ∂M in the interior of Λ incase (iii).In the special case that M = R , by the maximum principle [6, Corollary1.28] we know that all leaves of L are properly embedded (except whenΛ = ∂ R ). Therefore Lemma 2.6 implies the following reflection principlefor minimal lamination with free boundary. Lemma 5.4 (Lamination reflection principle) . If L is a minimal laminationof R with free boundary on ∂ R , then { Λ ∪ θ (Λ) : Λ ∈ L} is a minimallamination of R (in the sense of Definition 5.1). We will need the following convergence result. The proof will be post-poned until section 7.
Theorem 5.5.
Let ( M , g ) be a compact Riemannian -manifold with bound-ary ∂M (cid:54) = ∅ . If L i is a sequence of minimal laminations of M with freeboundary on ∂M of uniformly bounded curvature, i.e. there exists a con-stant C > such that sup {| A Λ | ( x ) : x ∈ Λ ∈ L i } ≤ C, then a subsequence of L i converges in the C α topology for any α < to aLipschitz lamination L with minimal leaves in M and free boundary on ∂M .Proof of Theorem 1.2. We follow the same contradiction argument as in theproof 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 laminations with bounded cur-vature [5, Proposition B.1] (for Type I convergence) and Theorem 5.5 (forType II convergence), a subsequence of blowups converges to • a minimal lamination ˜ L in T x M (cid:39) R , or • a minimal lamination L in H with free boundary on ∂H . In the second case, we can apply the lamination reflection principle (Lemma5.4) to obtain a minimal lamination ˜ L in T x M (cid:39) R . By the blowup as-sumption, we know that the origin 0 ∈ R is in the support of ˜ L , and thecurvature of the leaf Λ passing through 0 is exactly 1 at 0, i.e. | A Λ | (0) = 1.Now we analyze the structure of the minimal lamination ˜ L ⊂ R for bothcases. We refer to [12] for well-known terminologies for minimal laminations.If Λ ∈ ˜ L is an accumulating leaf, then either Λ or its double cover ˜Λ is acomplete, stable minimal surface in R , which must be an affine plane bythe Bernstein theorem in R (see [7, 8]). Therefore, the leaf Λ passingthrough 0 must be an isolated leaf. Since all the surfaces in the sequenceΣ (cid:48) i are stable with free boundary, the smooth convergence of Σ (cid:48) i to ˜ L or L and the reflection principle (Lemma 2.6) imply that Λ is a complete,stable, minimal surface in R . This again violates the Bernstein theorem as | A Λ | (0) = 1 by our construction. Therefore, we arrive at a contradictionand finish the proof of Theorem 1.2. (cid:3) Convergence of free boundary minimal submanifolds
In this section, we prove a general convergence result (Theorem 6.1) forfree boundary minimal submanifolds with uniformly bounded second fun-damental form. Note that this convergence result does not require stabilityand holds in any dimension and codimension.To facilitate our discussion, let us first review some basic properties of
Fermi coordinates . Let N n ⊂ M n +1 be an embedded hypersurface (withoutboundary) in the Riemannian manifold ( M, g ). We can assume that both N and M are complete. Fix a point p ∈ N , if we let ( x , · · · , x n ) be thegeodesic normal coordinates of N centered at p , and t = dist M ( · , N ) bethe signed distance function from N which is well-defined and smooth in aneighborhood of p inside M . Therefore, for r > Fermi coordinate chart , φ : B n +1 r (0) ⊂ T p M → U ⊂ M ( t, x , · · · , x n ) (cid:55)→ φ ( t, x , · · · , x n ) , such that U ∩ N = φ ( { t = 0 } ). Here, B n +1 r (0) is the open Euclidean ballof T p M ∼ = R n +1 of radius r > g in Fermi coordinates satisfy g tt = 1 and g x i t = 0for i = 1 , · · · , n .Let (Σ , ∂ Σ) ⊂ ( M, N ) be smooth embedded free boundary minimal k -dimensional submanifold, with 1 ≤ k ≤ n . Fix any p ∈ ∂ Σ ⊂ N , and let φ : B n +1 r (0) → U be a Fermi coordinate chart as above centered at p . Aftera rotation we can assume that T p ( ∂ Σ) = { x k = · · · = x n = 0 = t } ∼ = R k − . URVATURE ESTIMATES NEAR FREE BOUNDARY 17
Since Σ meets N orthogonally along ∂ Σ, after picking a choice on the signof t , the tangent half-space T p Σ is given by T p Σ = { x k = · · · = x n = 0 , t ≥ } ∼ = R k + . Hence, under the Fermi coordinates in a neighborhood of p , Σ can be writtenas a graph of u = ( u , · · · , u n +1 − k ) which is a R n +1 − k -valued function of( t, x (cid:48) ) = ( t, x , · · · , x k − ) in a domain of R k + , i.e. φ − (Σ) = { ( t, x (cid:48) , u ( t, x (cid:48) )) } ⊂ R n +1+ . Moreover, φ − ( ∂ Σ) is given by the same graph with t = 0. Since ∂∂t is a unitnormal vector field along N ∩ U , it is clear that the free boundary conditionalong ∂ Σ is equivalent to(6.1) ∂u (cid:96) ∂t (0 , x (cid:48) ) = 0 for (cid:96) = 1 , · · · , n + 1 − k .We now state the convergence result for free boundary minimal subman-ifolds with uniformly bounded area and the second fundamental form. Theorem 6.1.
Suppose we have a sequence (Σ j , ∂ Σ j ) ⊂ ( M, N ) of im-mersed free boundary minimal k -dimensional submanifolds, where ≤ k ≤ n , with uniformly bounded area and second fundamental form, i.e. thereexist positive constants C , C > such that Area(Σ j ) ≤ C and sup Σ j | A Σ j | ≤ C for all j , then after passing to a subsequence, (Σ j , ∂ Σ j ) converges smoothlyand locally uniformly to (Σ ∞ , ∂ Σ ∞ ) ⊂ ( M, N ) which is a smooth immersedfree boundary minimal k -dimensional submanifold.Proof. The convergence away from N follows from the classical convergenceresults. By the second fundamental form bound, we can cover N by balls (ofa uniform size) under Fermi coordinates centered at p ∈ N such that each Σ j can be written as graphs over some domain of T p Σ j with uniformly boundedgradient (see [6, § j and each sheet is a graph overa k -dimensional subspace of T p M or a k -dimensional half-space orthogonalto T p N . The first case again follows from the classical interior convergenceresult. The second case follows from standard elliptic PDE theory withNeumann boundary conditions (6.1) (see [1] for example). (cid:3) Convergence of free boundary minimal lamination
Finally, we give the proof of Theorem 5.5 which was used in section 5.
Proof of Theorem 5.5.
For simplicity we will assume that each lamination L i has finitely many leaves where the number of leaves may depend on i ;this will suffice for our application. For any interior point x ∈ M \ ∂M , the argument used in the proof of [5, Proposition B.1] implies the convergencein a small neighborhood of x in M \ ∂M . Hence, we only need to deal withthe convergence near a boundary point x ∈ ∂M .Fix p ∈ ∂M and let N = ∂M . The theorem will follow once we con-struct uniform coordinate charts in a small neighborhood of p in the Fermicoordinate system as in section 6. Let ϕ be a Fermi coordinate chart in arelatively open neighborhood U of p in M , i.e., ϕ : U ⊂ M → (cid:101) U ⊆ R , such that ϕ ( p ) = 0 and ϕ ( N ∩ U ) = { x = 0 } ∩ (cid:101) U . Here, ( x , x , x ) arethe local Fermi coordinate system centered at p (i.e. t = x ). Suppose that B +4 r ⊂ (cid:101) U for some small r to be chosen later, where B +4 r = B r ∩ { x ≥ } denotes the half ball in R with radius 4 r centered at the origin.Next, we will construct uniform coordinate charts on ϕ − ( B + r ). Note thatfor each i and every Λ ∈ L i , we have sup Λ | A Λ | ≤ C. We may choose r sufficiently small so that Cr is as small as we wish. Then for each fixed i , (cid:91) Λ ∈L i ϕ (Λ ∩ U ) ∩ B +4 r gives a finite number of disconnected surfaces with bounded curvature inthe Fermi coordinate system.Since the lamination has uniformly bounded curvature, by the tilt esti-mates as in the proof of [6, Lemma 2.11], there exists a constant δ > L i , we have the following two cases: (i) none ofthe leaves of L i meets ∂ R in B + δr (except possibly for one leaf touching ∂ R tangentially at some points); (ii) there exists a leaf of L i meeting ∂ R along some non-empty free boundary. For case (i), we can construct uniformcoordinate charts as in the proof of [5, Proposition B.1] in a neighborhood ofthe larger manifold (cid:102) M . For case (ii), we claim that in B +2 δr , all leaves of L i which intersect B + δr must meet ∂ R along some non-empty free boundary;otherwise, the tilt estimates will imply that two leaves intersect somewherein B + r which contradicts the assumption that all leaves are disjoint. Notethat the tilt estimates in [6, Lemma 2.11] only use the uniform curvaturebound of leaves in L i , but not the minimal surface equations.Now, we focus on case (ii). For simplicity, we use r to denote δr . Thefree boundary condition and the choice of Fermi coordinates imply that thesesurfaces meet ∂ R orthogonally in the Euclidean metric. Going to a furthersubsequence (possibly with r even smaller), for fixed i , every sheet of (cid:91) Λ ∈L i ϕ (Λ ∩ U ) ∩ B +2 r , which intersects B + r is a graph with small gradient over a subset of certainfixed plane perpendicular to ∂ R (which can be chosen as R ×{ } := { x = URVATURE ESTIMATES NEAR FREE BOUNDARY 19 } after a rotation keeping ∂ R fixed as a set) containing a half ball of radius r (see [6, Lemma 2.4]).We will show that in a concentric half ball of smaller radius in B +2 r , thesequence of laminations converges in the C α topology to a lamination for any α <
1. The coordinate chart Φ required by the definition of a lamination willbe given by the Arzela-Ascoli theorem as a limit of a sequence of bi-Lipschitzmaps Φ i : B +2 r → R with bounded bi-Lipschitz constants, and Φ will be defined on a slightlysmaller concentric half ball B + sr for some s > i fixedΦ i (cid:0) B + sr ∩ ϕ ( ∪ Λ ∈L i Λ ∩ U ) (cid:1) is the union of subsets of planes which are each parallel to R × { } ⊆ R . Set the map Φ i by lettingΦ − i ( y , y , y ) = ( y , y , φ i ( y , y , y )) , where φ i is defined as follows: order the sheets of B +2 r ∩ ϕ ( ∪ Λ ∈L i Λ ∩ U ) asΛ i,k for k = 1 , . . . by increasing values of x and let Λ i,k be the graph ofthe function f i,k over (part of) the R × { } plane. In the following we onlyneed to consider those sheets Λ i,k where Λ i,k ∩ B + r (cid:54) = ∅ , since we eventuallywill work on a much smaller concentric half ball. The domain of such f i,k contains the half ball of radius r centered at the origin of the R ×{ } plane.Again as Cr can be chosen small enough, we can assume that |∇ f i,k | are assmall as we want. Moreover, the free boundary condition satisfied by Λ i,k isequivalent to the Neumann boundary condition:(7.1) ∂f i,k (0 , · ) ∂x = 0 . Set w i,k = f i,k +1 − f i,k . In the following, ∆, ∇ , and div will be withrespect to the Euclidean metric on R × { } . By a standard computation(cf. [6, Chapter 7] or [20, (7)]), we have(7.2) div(( a + Id ) ∇ w i,k ) + b ∇ w i,k + cw i,k = 0 , and ∂w i,k (0 , · ) ∂x = 0 , where a is a matrix-valued function, b is a vector-valued function, and c issimply a real-valued function.Note that a , b , and c depend on i , but the norms of a, b, c can be madeuniformly small if Cr is small enough and if we rescale our ambient manifoldby a large factor. By (7.2), and the Harnack inequality (see [9, 8.20] and [1,Section 6]) applied to the positive function w i,k gives(7.3) sup B +2 sr w i,k ≤ C inf B +2 sr w i,k , where C depends only on the norms of a, b and c . Here, B + t is the half ballin R × { } with radius t and center 0. Set M i,k = f i,k (0 , { ( y , y , y ) ∈ B + r × [ M i,k , M i,k +1 ] } , define the function φ i by φ i ( y , y , y ) = f i,k ( y , y ) + y − M i,k M i,k +1 − M i,k w i,k ( y , y ) . Hence, Φ − i (cid:0) y , y , f i,k (0 , (cid:1) = (cid:0) y , y , f i,k ( y , y ) (cid:1) ;that is, Φ i maps Λ i,k to a subset of the plane R × { f i,k (0 , } .Note that φ i (0 , ,
0) = 0. Moreover, we have(7.4) ∇ φ i = ∇ f i,k + y − M i,k M i,k +1 − M i,k ∇ w i,k + w i,k M i,k +1 − M i,k ∂∂y . By (7.3) and (7.4), we know that for each i the map Φ i restricted to B + sr ⊆ R is bi-Lipschitz with uniformly bounded bi-Lipschitz constant.By the Arzela-Ascoli theorem, a subsequence of Φ i converges in the C α topology for any α < N (for the second case). (cid:3) References
1. S. Agmon, A. Douglis, and L. Nirenberg,
Estimates near the boundary for solutions ofelliptic partial differential equations satisfying general boundary conditions. I , Comm.Pure Appl. Math. (1959), 623–727. MR 01253072. William K. Allard, On the first variation of a varifold: boundary behavior , Ann. ofMath. (2) (1975), 418–446. MR 03975203. Simon Brendle,
A sharp bound for the area of minimal surfaces in the unit ball , Geom.Funct. Anal. (2012), no. 3, 621–626. MR 29726034. Adrian Butscher, Deformations of minimal Lagrangian submanifolds with boundary ,Proc. Amer. Math. Soc. (2003), no. 6, 1953–1964 (electronic). MR 19552865. Tobias Holck Colding and William P. Minicozzi, II,
The space of embedded minimalsurfaces of fixed genus in a 3-manifold. IV. Locally simply connected , Ann. of Math.(2) (2004), no. 2, 573–615. MR 2123933 (2006e:53013)6. ,
A course in minimal surfaces , Graduate Studies in Mathematics, vol. 121,American Mathematical Society, Providence, RI, 2011. MR 27801407. M. do Carmo and C. K. Peng,
Stable complete minimal surfaces in R are planes ,Bull. Amer. Math. Soc. (N.S.) (1979), no. 6, 903–906. MR 5463148. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfacesin 3-manifolds of nonnegative scalar curvature , Comm. Pure and Appl. Math. (1980), 199–211.9. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of secondorder , Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998edition. MR 181436410. Michael Gr¨uter and J¨urgen Jost,
Allard type regularity results for varifolds with freeboundaries , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1986), no. 1, 129–169.MR 863638 URVATURE ESTIMATES NEAR FREE BOUNDARY 21
11. Alexei Kovalev and Jason D. Lotay,
Deformations of compact coassociative 4-foldswith boundary , J. Geom. Phys. (2009), no. 1, 63–73. MR 247926312. Haozhao Li and Xin Zhou, Existence of minimal surfaces of arbitrarily large Morseindex , Calc. Var. Partial Differential Equations (2016), no. 3, Paper No. 64, 12.MR 350903813. Martin Man-chun Li and Xin Zhou, Min-max theory for free boundary minimal hy-persurfaces I: regularity theory , arXiv: 1611.02612.14. Jon T. Pitts,
Existence and regularity of minimal surfaces on Riemannian manifolds ,Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; Universityof Tokyo Press, Tokyo, 1981. MR 62602715. Antonio Ros,
One-sided complete stable minimal surfaces , J. Differential Geometry (2006), 69–92.16. R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces ,Acta Math. (1975), no. 3-4, 275–288. MR 042326317. Richard Schoen,
Estimates for stable minimal surfaces in three-dimensional mani-folds. seminar on minimal submanifolds , vol. 103, Ann. of Math. Stud., no. 111-126,Princeton Univ. Press, Princeton, NJ, 1983.18. Richard Schoen and Leon Simon,
Regularity of stable minimal hypersurfaces , Comm.Pure Appl. Math. (1981), no. 6, 741–797. MR 63428519. Leon Simon, Lectures on geometric measure theory , Proceedings of the Centre forMathematical Analysis, Australian National University, vol. 3, Australian NationalUniversity, Centre for Mathematical Analysis, Canberra, 1983. MR 75641720. ,
A strict maximum principle for area minimizing hypersurfaces , J. DifferentialGeom. (1987), no. 2, 327–335. MR 90639421. Alexander Volkmann, A monotonicity formula for free boundary surfaces with respectto the unit ball , Comm. Anal. Geom. (2016), no. 1, 195–221. MR 3514558 Department of Mathematics, University of California Santa Barbara, SantaBarbara, CA 93106, USA
E-mail address : [email protected] Department of Mathematics, The Chinese University of Hong Kong, Shatin,N.T., Hong Kong
E-mail address : [email protected] Department of Mathematics, South Hall 6501, University of CaliforniaSanta Barbara, Santa Barbara, CA 93106, USA
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