Curvature estimates for stable minimal surfaces with a common free boundary
CCURVATURE ESTIMATES FOR STABLE MINIMAL SURFACESWITH A COMMON FREE BOUNDARY
GAOMING WANG
Abstract.
The minimal surfaces meeting in triples with equal angles along acommon boundary naturally arise from soap films and other physical phenom-enon. They are also the natural extension of the usual minimal surface. Inthis paper, we consider the multiple junction surface and show the BernsteinTheorem still holds for stable multiple junction surface in some special case.The key part is to derive the L p estimate of the curvature for multiple junctionsurface. Introduction
In the paper of Schoen, Simon, Yau [1], they’d showed the L p curvature esti-mate for the minimal hypersurfaces. As a corollary, they could get the generalizedBernstein’s theorem. That is, the only stable immersed hypersurface Σ n (cid:44) → R n +1 (dim Σ n = n and n ≤ ) with area growth condition (i.e. H n ( B Σ R ) ≤ CR n for all theintrinsic ball B Σ with radius R ) is a hyperplane. In [2], Colding, Minicozzi showedthe stable and -sided, simply connected minimal surface in R has quadratic areagrowth. So the only stable, complete, 2-sided minimal surface in R is a plane.Thus, considering the triple junctions appearing in the nature phenomenon such assoap film, we have the following nature problem. Problem 1.1.
If three 2-sided minimal surfaces-with-boundary meet at the sameboundary and each two of them meet at exactly degrees along Γ . Suppose theyare complete under the distance function and stable in some suitable variations, isit true that each of them is flat? Clearly, we need the careful definition of the surface, completeness and stability.We will give the detailed definitions in Section 2.In this paper, we will give the partial result to this problem like Schoen, Simon,Yau’s result. Instead of triple junctions, we consider arbitrary number of surfacesmeet at the same boundary.
Theorem 1.2.
Suppose M = ( θ Σ , · · · , θ q Σ q ; Γ) is the (orientable) minimal mul-tiple junction surface in R . We assume M is complete, stable and has quadraticarea growth. Furthermore, we assume Γ is compact and the angles between Σ i , Σ j keep same along Γ for ≤ i, j ≤ q . Then each Σ i is flat. The terminology will be explained in Section 2.The special case is the triple junction surface, M = (Σ , Σ , Σ ; Γ) with Γ com-pact. Note the angles between Σ i , Σ j are π for all i (cid:54) = j . So from the abovetheorem, we know the Y -shaped catenoid is non-stable in our sense.Another case is the Y -shaped bent helicoid. Like the usual bent helicoid, onecan construct Y -shaped bent helicoid by the classical Björling’s formula(see [3] forexample). One can choose three unit normal vector fields making equal angles witheach other instead of one along a unit circle in the construction. It has a circle asthe triple junction and hence it is not stable. Date : February 22, 2021. a r X i v : . [ m a t h . DG ] F e b GAOMING WANG Σ Σ Σ Σ Σ Σ Figure 1.
Two kinds of Y -shaped catenoid Figure 2. Y -shaped bent helicoid and usual helicoidFor the case if Γ is a straight line, we also have similar result. Theorem 1.3.
Suppose M has the same condition with Theorem 1.1 except Γ beinga straight line instead of being compact. The each Σ i is flat. Note that this theorem is not enough to show the Y -shaped helicoid is non-stablesince it does not have quadratic area growth.The key step proving the above theorems is the following curvature estimate forminimal multiple junction surface. Theorem 1.4.
Suppose M is the minimal multiple junction surface. We assume M is stable. Then for the smooth function φ i defined on Σ i with compact supportand satisfying compatible condition along Γ , i.e. sign ( φ i ) | A i | p − | φ i | p will be theprojection of a smooth vector field along Γ to the normal direction of Σ i . Then wehave q (cid:88) i =1 (cid:90) Σ i θ i | φ i | p | A i | p ≤ q (cid:88) i =1 θ i (cid:20)(cid:90) Σ i C |∇ φ i | | A i | p − | φ i | p − + (cid:90) Γ (cid:18) p − | τ i (log | A i | ) | − (cid:104) H Γ , τ i (cid:105) (cid:19) | A i | p − | φ i | p (cid:21) (1.1) for p ∈ (1 , ) , where H Γ is the curvature vector of Γ , τ i is the outer conormal of Σ i along Γ . The constant C = C ( M ) doesn’t rely on p . The compatible condition will make sure the variation is well-defined on M , see(2.2) for details. Note that the integration of the term | A i | p − | φ i | p τ i ( | A i | ) is stillwell defined for | A i | = 0 as we’ll explained later on.For the case of triple junction, i.e. q = 3 and θ = θ = θ = 1 , this condition isequivalent to the following identity (cid:88) i =1 sign ( φ i ) | A i | p − | φ i | p = 0 URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY3
The proof of (1.1) is essentially following Schoen, Simon, Yau’s proof [1]. Beforethat, we need to calculate the second variation formula to get the following stabilityoperator.(1.2) q (cid:88) i =1 (cid:90) Σ i θ i (cid:16) |∇ Σ i φ i | − | A i | φ i (cid:17) − (cid:90) Γ θ i φ i H Γ · τ i with φ i satisfying compatible condition (2.2).After getting curvature estimate, we can choose a suitable function. The trickpart is we need our functions to satisfy the compatible conditions. So near Γ , φ i should satisfy some compatible conditions and the gradient φ i cannot vanish near Γ . This is why we need Γ is compact or Γ is a straight line. So we can choose p close to 1 to control the term near Γ . For the part that far from Γ , we can choose φ i like the standard cutoff function in a large ball. After choosing a suitable function,we can deduce the curvature needs to vanish everywhere.2. Minimal surfaces with multiple junction
In this section, we will fix some notations and give the definition of the minimalmultiple junction surface and several related concepts.For q ∈ N , we suppose Σ , · · · , Σ q are all smooth 2-dimensional manifolds withboundary ∂ Σ i and Γ is a smooth 1-dimensional manifold. We only consider thecase that each Σ i is orientable so we have the well defined unit normal vector fieldon Σ i when immersing into R . For each i , we suppose there is a diffeomorphism p i : Γ → ∂ Σ i . X · Y denotes the standard inner product in R . D X Y denotes the standard coderivative in R . B r ( x ) denotes the open ball centered at x with radius r in R .For a immersed manifold Σ into R , we use T Σ to denote the tangent bundle of Σ , and N Σ denote the normal bundle of Σ in R .For Σ i (cid:44) → R immersed in R , we use the following notations. A i denote the second fundamental form on Σ i . | A i | , the norm of second fundamental form on Σ i . ν i denotes the unit normal vector field on Σ i . τ i denotes the outer conormal of ∂ Σ i on Σ i . That is, τ i is the smooth unit normalvector field on ∂ Σ i such that τ i ∈ C ∞ (Γ , T Σ i ) and τ i pointing outside of Σ i .sign ( x ) denotes the sign function, i.e. sign ( x ) = 1 for x > , sign (0) = 0 andsign ( x ) = − for x < . H i (resp. H Γ ) denotes the mean curvature vector of Σ i (resp. Γ ).2.1. Definition of multiple junction surfaces.Definition 2.1.
We say M = (Σ , · · · , Σ q ; Γ) is an intrinsic multiple junctionsurface if it is a quotient space ∪ qi =1 Σ i / ∼ where the equivalent relation is definedas the following, x ∼ y if and only if x = y or x ∈ ∂ Σ i , y ∈ ∂ Σ j for some ≤ i, j ≤ q and x = p i ◦ p − j ( y ) . Remark.
We can define M as a topological space with coordinate charts like thedefinition of smooth manifold. Definition 2.2.
We say M = (Σ , · · · , Σ q ; Γ) is a multiple junction surface in R if (Σ , · · · , Σ q ; Γ) is an intrinsic multiple junction surface and there is a map ϕ : M → R such that the restriction of ϕ on Σ i is a smooth immersion for each ≤ i ≤ q .We call the map ϕ smooth immersion for M . GAOMING WANG
Note that we have a nature metric on each Σ i for ≤ i ≤ q by pulling back themetric on R .In general, we will consider the multiple junction surface M = (Σ , · · · , Σ q ; Γ) with constant density θ , · · · , θ q > such that we have constant density function θ i on the surface Σ i . We will write this surface as M = ( θ Σ , · · · , θ q Σ q ; Γ) . So theassociated 2-varifold of M has the form V M := q (cid:88) i =1 θ i | Σ i | . Here, | Σ i | denotes the multiplicity one varifold associated with the surface Σ i . Notethat we do not require θ i to be the integers. Definition 2.3.
We say a multiple junction surface M = ( θ Σ , · · · , θ q Σ q ; Γ) isminimal if each Σ i is a smooth minimal immersion and on Γ , we have q (cid:88) i =1 θ i τ i = 0 . Remark.
By the regularity of B. Krummel [4], suppose a stationary integral 2-varifold V has the form V = q (cid:88) i =1 θ i | Σ k | for distinct C ,µ embedded hypersurfaces-with-boundary Σ , · · · , Σ q with a com-mon boundary Γ for some < µ < . Then for any Z ∈ Γ , if T Z Σ i are not the sameplane in R , then we can find a neighborhood O Z of Z such that Σ i is smooth and Γ is a smooth curve in O Z . Indeed, they are all analytic since R is a real analyticmanifold.Moreover, by the regularity of cylindrical tangent cones by L. Simon [5], if astationary integral 2-varifold in U has density at some point Z ∈ U , then near Z , M is the varifold associated with three C ,µ minimal surface with a commonboundary Γ and Γ is still a C ,µ curve for some < µ < . So at least for the triplejunction, we can assume much weaker condition on the above definition.For each Σ i , we can define the intrinsic distance function d i ( x, y ) for x, y ∈ Σ i ,which is the length of the shortest geodetic jointing x, y on Σ i .So we can define a global distance function d ( x, y ) for x ∈ Σ i , y ∈ Σ j by d ( x, y ) := inf (cid:40) l − (cid:88) k =0 d i k ( x k , x k +1 ) : x = x, x l +1 = y, x , · · · , x l ∈ Γ ,i = i, i l − = j, ≤ i , · · · , i l − ≤ q, for l ∈ N (cid:41) . Hence, we use B Mr ( x ) = { y ∈ M : d ( x, y ) < r } to denote the intrinsic ball on M .Now we can define the distance function with respect to Γ as d Γ ( x ) = inf y ∈ Γ d ( x, y ) for x ∈ M. Definition 2.4.
We say a multiple junction surface M is complete if it is completein the distance function d ( · , · ) . That is, every Cauchy sequence converges to somepoint in M under this distance function. URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY5
Definition of functional spaces on triple junction surfaces.
From nowon, we will always assume M = ( θ Σ , · · · , θ q Σ q ; Γ) is a complete minimal multiplejunction surface in R .When we consider the variation on M , we will need to consider a kind of vectorfield on M . So we have the following definition. Definition 2.5.
We say a map X ( x ) : M → T x R (cid:39) R is a C k vector field on M if each X | Σ i is a C k vector field on Σ i for ≤ i ≤ q . We write this vector fieldspace as C k ( M ; T R ) .Note that we do not require the vector field can be jointed smoothly cross thejunction. For example, let M = ( H , H ; Γ) with H , H the opposed two half planesin R and Γ the straight line in R . Then as a immersion, M can be regarded asa smooth plane in R but the smooth vector field on M may not smooth on thisplane.Let’s consider the space of functions on M . The nature definition is to considerthe function on M , which write as f : M → R and say it is C k if the restriction oneach Σ i is C k up to boundary.Somehow this function space is not big enough to contain the function we areinterested. For example, give a vector field V ∈ C k ( M, T R ) , the function definedby V · ν i for x ∈ Σ i is not a C k function defined above. Actually it isn’t well-definedon M since on Γ , the value will depend on i . So we define some large function spacesas following. Definition 2.6.
We say a function f ( x ) : ∪ qi =1 Σ i → R is in a Sobolev space W k,p ( M ) for ≤ p ≤ ∞ if each restriction f | Σ i is in W k,p (Σ i ) for each i .Similarly, we can define the L p space as L p ( M ) and continuous function space C k ( M ) . Usually, we will write H k ( M ) = W k, ( M ) to denote it as Hilbert space.By our definition, we do not impose any condition along Γ for f ∈ C k ( M ) . Ingeneral, we still wish our function can also be extended to a suitable vector fieldon M at least. So we say f ∈ C k ( M ) satisfies compatible condition if there existsa C k vector field W along Γ (i.e. W ∈ C k (Γ , T R ) ), such that(2.1) f i ( x ) = W ( p − i ( x )) · ν i ( x ) for x ∈ ∂ Σ i , ≤ i ≤ q where f i = f | Σ i .Note that by Trace Theorem, if f ∈ W ,p ( M ) for some ≤ p ≤ ∞ , then for any ≤ i ≤ q , the function f i can be restricted to the boundary ∂ Σ i in the L p ( ∂ Σ i ) sense.So we can say f ∈ W ,p ( M ) satisfies compatible condition if there is a L p loc (Γ) vector field W along Γ , such that(2.2) f i | ∂ Σ i ( x ) = W ( p − i ( x )) · ν i ( x ) for H -a.e. x ∈ ∂ Σ i , ≤ i ≤ q. Sometime we will write (2.1),(2.2) as f i = W · ν i for short.Clearly, the function defined by V · ν i is in C k ( M ) for V ∈ C k ( M, T R ) andsatisfies (2.1). Conversely, for any f ∈ C k ( M ) satisfying (2.1), by definition wehave W ∈ C k (Γ , T R ) , so f i = W · ν i . For each ≤ i ≤ q , we can extend W (cid:62) on the whole Σ i to ˜ V i such that ˜ V i is a C k tangential vector field on Σ i . This isbecause W (cid:62) is C k on Γ and Γ is smooth on Σ i . So we can define V i = ˜ V i + f i ν i ,which is a C k vector field on Σ i . So the vector field V defined by V = V i on Σ i isin C k ( M, T R ) and satisfies V · ν i = f i . Remark.
For the triple junction surface, the compatible condition has a simpleform. For f ∈ W ,p ( M ) , f satisfies compatible condition if and only if f + f + f = 0 H -a.e. on Γ . GAOMING WANG
Remark.
All the definitions in this section can extend to arbitrary ambient man-ifolds with arbitrary dimension and codimension.3.
First and second variation of M Now we can consider the variation of M . We say M t = ( θ Σ t , · · · , θ q Σ qt ; Γ t ) , t ∈ ( − ε, ε ) (considered as immersion) is a C k variation of M if each Σ it is a C k variationof Σ i up to boundary and Γ t is a C k variation of Γ . Of course, we can write thisvariation as one-parameter family of immersion ϕ t ( x ) := ϕ ( t, x ) : ( − ε, ε ) × M → R such that for each t , M t = ϕ t ( M ) is a multiple junction surface and restrict on each Σ i the variation ϕ t is C k .For each C k variation ϕ t : M → R , there is an associated vector field V ( x ) : M → T x R , which is C k on Σ i for each ≤ i ≤ q .Let U ∈ M be a open subset in M such that U is compact. Suppose we havea C k variation for M with associated vector field V ( x ) = ∂ϕ t ( x ) ∂t with compactsupport in U . We can define the first variation of the area of M in U , which isgiven by dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 | ϕ t ( U ) | = (cid:90) M ∩ U div T x M V ( x ) d (cid:107) V M (cid:107) ( x )= q (cid:88) i =1 (cid:90) Σ i ∩ U div T x Σ i V ( x ) θ i dµ Σ i ( x )= q (cid:88) i =1 − (cid:90) Σ i ∩ U V · H i θ i dµ Σ i + q (cid:88) i =1 (cid:90) Γ V · τ i θ i dµ Γ where div P V = D e V · e + D e V · e for any orthonormal basis e , e of the plane P . The µ Σ i is the area measure on Σ i . (cid:107) V M (cid:107) is the weight measure of V M .We say M is stationary in U if for any such variation, we have dd t | ϕ t ( U ) | = 0 .Note that every C k vector field on M will give a C k variation of M . So M isstationary in U if and only if H i = 0 on each Σ i ∩ U and on Γ ∩ U , we have q (cid:88) i =1 θ i τ i = 0 . This is precisely the condition that we define the minimal multiple junction surface.Now we can consider the second variation of area for minimal multiple junctionsurface.
Definition 3.1.
We say a minimal triple junction surface M is stable in U whoseclosure is compact if for every variation ϕ t of M in U , we haved d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 | ϕ t ( M ∩ U ) | ≥ . So we say M is stable if for every U with compact closure, we always have d d t (cid:12)(cid:12)(cid:12) t =0 | ϕ t ( M ∩ U ) | ≥ for any variation ϕ t in U .The remaining part of this section is to deduce the stability operator (1.2). Theorem 3.2. If M is a stable complete minimal multiple junction surface in R .Then for any φ ∈ C k ( M ) satisfying (2.1) with compact support, we have (3.1) q (cid:88) i =1 (cid:90) Σ i (cid:16) |∇ Σ i φ i | − | A i | φ i (cid:17) θ i dµ Σ i − (cid:90) Γ φ i H Γ · τ i θ i dµ Γ ≥ URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY7 where, ∇ Σ i φ i denotes the gradient on Σ i and | A i | denotes the norm of the secondfundamental form on Σ i . H Γ means the curvature vector of the curve Γ . Moreover,it holds even for φ ∈ H ( M ) satisfying (2.2) with compact support.Proof. Let φ ∈ C k ( M ) . Since it satisfies compatible condition, we can find V ∈ C k ( M, T R ) such that φ i = V · ν i . So there is a variation ϕ t with compact supportassociated with vector field V , i.e. V = dd t (cid:12)(cid:12) t =0 ϕ t Suppose ϕ t is supported in U . So by the first variation formula, we havedd t | ϕ t ( M ∩ U ) | = q (cid:88) i =1 − (cid:90) Σ it V t · ν it H Σ it θ i dµ Σ it + q (cid:88) i =1 (cid:90) Γ t V t · τ it θ i dµ Γ t where H Σ it = H Σ it · ν it and V t = dd t ϕ t .So after taking derivative with respect to t on the first variation formula, wehave d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 | ϕ t ( M ∩ U ) | = q (cid:88) i =1 − (cid:90) Σ i V · ν i (cid:18) dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 H Σ it (cid:19) θ i dµ Σ i − q (cid:88) i =1 (cid:90) Σ i H Σ i θ i dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( V t · ν it dµ Σ it )+ q (cid:88) i =1 (cid:90) Γ V · dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 τ it θ i dµ Γ t + q (cid:88) i =1 (cid:90) Γ θ i τ i · dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( V t dµ Γ t ) (3.2)Note that by stationary condition, we know H Σ i = 0 and (cid:80) qi =1 θ i τ i = 0 along Γ , so the second and forth terms in (3.2) vanish. Moreover, we’ve the well knownformula (cf. [6])(3.3) dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 H Σ it = ∆ Σ i φ i + | A i | φ i So actually, we only need to compute V · dd t (cid:12)(cid:12) t =0 τ it . For simplicity, we use ( · ) (cid:48) todenote dd t (cid:12)(cid:12) t =0 ( · ) .Before computing τ (cid:48) i , we need to get ν (cid:48) i . Let e , e be the orthonormal frame of T x Σ i for some x ∈ Σ i . Let e it = dϕ t ( e i ) .We can decompose V = φ i ν i + W i on Σ i with W i tangential to Σ i . Note that [ e it , V t ] = 0 , we have ν (cid:48) i · e i = − ν i · e (cid:48) i = − ν i · D e i V = − e i ( φ i ) − ν i · D e i W = − e i ( φ i ) − A Σ i ( e i , W ) Suppose η is the unit tangential vector field on Γ and also write η t = dϕ t ( η ) .This time we decompose V as V = φ i ν i + f i τ i + gη where f i = V · τ i , g = V · η .Similarly with ν (cid:48) i , we have η (cid:48) · τ i = τ i · D η V = φ i τ i · D η ν i + η ( f i ) + gτ i · D η η = − φ i A Σ i ( τ i , η ) + η ( f i ) + g H Γ · τ i Hence, τ (cid:48) i · V = ( τ (cid:48) i · ν i ) φ i + ( τ (cid:48) i · η ) g = − ( τ i · ν (cid:48) i ) φ i − ( τ i · η (cid:48) ) g = τ i ( φ i ) φ i + φ i A Σ i ( τ i , W ) + φ i gA Σ i ( τ i , η ) − gη ( f i ) − g H Γ · τ i = τ i ( φ i ) φ i + φ i f i A i ( τ i , τ i ) + 2 φ i gA i ( τ i , η ) − gη ( f i ) − g H Γ · τ i GAOMING WANG
Note that we can take V which is a normal vector field on when restricting on Γ . In this case, we have g = 0 on Γ , so we get q (cid:88) i =1 θ i τ (cid:48) i · V = q (cid:88) i =1 θ i [ τ i ( φ i ) φ i + φ i f i A i ( τ i , τ i )] (3.4)Beside, if we do not assume V is normal to Γ , we can still get (3.4) by notingthe following identity φ i A i ( τ i , η ) = φ i D η τ i · ν i = D η τ i · V − D η τ i · ( f i τ i ) − D η τ i · ( gη ) So after taking sum over i with density θ i , we can use minimal condition (cid:80) qi =1 τ i θ i =0 to get same result as (3.4).To further processed, we use each Σ i is a minimal surface and get φ i f i A i ( τ i , τ i ) = − φ i f i A i ( η, η ) = − f i D η η · ( φ i ν i )= − f i H Γ · V + f i H Γ · τ i = − f i H Γ · V + ( | V | − φ i − g ) H Γ · τ i After taking sum of i , we have q (cid:88) i =1 θ i φ i f i A i ( τ i , τ i ) = 0 + 0 − q (cid:88) i =1 θ i φ i H Γ · τ i − q (cid:88) i =1 θ i φ i H Γ · τ i Hence, combining (3.2), (3.3), (3.4), we get the second variation formula asd d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 | ϕ t ( M ∩ U ) | = q (cid:88) i =1 (cid:90) Σ i (cid:16) − φ i ∆ Σ i φ i − | A i | φ i (cid:17) θ i dµ Σ i + q (cid:88) i =1 (cid:90) Γ (cid:0) τ i ( φ i ) φ i − φ i H Γ · τ i (cid:1) θ i dµ Γ = q (cid:88) i =1 (cid:18)(cid:90) Σ i (cid:16) |∇ Σ i φ i | − | A i | φ i (cid:17) θ i dµ Σ i − (cid:90) Γ φ i H Γ · τ i θ i dµ Γ (cid:19) So this theorem is followed by the definition of the stable.Note that we can approach the function in H ( M ) locally by smooth functions,and the compatible condition will keep hold in the trace sense. So the aboveinequality still holds when φ ∈ H ( M ) satisfying (2.2) with compact support. (cid:3) Remark.
The proof of Theorem 3.2 can be extend to higher dimensional multiplejunction hypersurfaces in an arbitrary complete ambient manifold N directly. Thestability operator will have the form q (cid:88) i =1 (cid:90) Σ i (cid:104) |∇ Σ i φ i | − Ric N ( ν i ) φ i − | A i | φ i (cid:105) θ i − (cid:90) Γ φ i (cid:104) H Γ , τ i (cid:105) θ i . Functions with finite orders
Before giving the proof of L p estimate, we need to consider some special func-tional spaces on M containing | A i | and test functions we’re interested in. Specifi-cally, at least we want to show τ i (log | A i | ) | A i | p − are locally integrable for each p > .Let’s fix a surface Σ (cid:44) → R with smooth boundary Γ . We will assume < α < ∞ . Definition 4.1.
We say a non-negative function g ( x ) on Σ has smooth order α near x if there is a conformal coordinate chart ϕ ( z ) : V → U ⊂ Σ with ∈ V ⊂ C and ϕ (0) = x such that g ( z ) := g ( ϕ ( z )) has form g ( z ) = h ( z ) | z | α URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY9 where h ( z ) is positive and smooth in V .Here, the conformal coordinate chart is the coordinate chart that metric near x has form λ ( z ) | dz | . We allow x to be on Γ so that V is a domain with smoothboundary in C . Definition 4.2.
We call a non-negative function g has smooth finite order on Σ ,if there exists a discrete subset P ⊂ Σ such that g is smooth and positive on Σ \ P and g has smooth order α x near x for each x ∈ P . We write this function space as ˜ C + (Σ) for conveniences.Similarly, we say a non-negative function g ( t ) on Γ has smooth order α near x if g ( t ) can be written as g ( t ) = | t | α h ( t ) for some smooth positive function h ( t ) undersome arc length parametrization with g (0) = x . So we can define smooth finiteorder function space ˜ C + (Γ) on Γ which contains the functions smooth outside adiscrete set P and has smooth order near each point of P .We have the following lemma for the relation of these two spaces. Lemma 4.3.
Let Σ be a two dimensional analytic Riemannian manifold withsmooth boundary ∂ Σ . For any ˜ g ∈ ˜ C + (Γ) , there is an extension of ˜ g denotedby g such that g ∈ ˜ C + (Σ) . Moreover, we can require g is positive on Σ \ ∂ Σ .Conversely, for any g ∈ ˜ C + ( ∂ Σ) , the restriction of g on Γ lies in ˜ C + (Γ) .Proof. Let’s write P = { x ∈ Γ : ˜ g ( x ) = 0 } . We can just focus on the extension neareach x ∈ P since ∂ Σ is smooth on Σ and positive smooth function can be easilyextended from ∂ Σ to Σ locally and keep positivity. Then we can use partition ofunity to get a global extension.Fix x ∈ P , we choose a conformal coordinate ϕ ( z ) : V → U where V, U are allhomeomorphic to a half disk such that ϕ (0) = x . WOLG, we assume the metrichas form λ ( z ) dzdz with λ (0) = 1 in this chart. Moreover, we can assume U issmall enough such that ∂ Σ ∩ U has an arc length parametrization γ ( t ) : ( a, b ) → Γ with γ (0) = x .Let’s us consider the function f ( t ) := | t || ϕ − ◦ γ ( t ) | defined in ( − ε (cid:48) , ε (cid:48) ) \ for somesmall ε (cid:48) where | z | is the usual absolute value in C with respect to this coordinatechart. We want to show that, by define f (0) = 1 , we can get a smooth function f on ( − ε (cid:48)(cid:48) , ε (cid:48)(cid:48) ) .Note that the map ψ ( t ) := ϕ − ◦ γ ( t ) is smooth from ( − ε (cid:48) , ε (cid:48) ) to V ⊂ C with ψ (0) = 0 , so we can expand ψ ( t ) as ψ ( t ) = ψ (cid:48) (0) t + ψ ( t ) t for some smooth map ψ near 0.So we have f ( t ) = | t || ψ ( t ) | = 1 | ψ (cid:48) (0) + ψ ( t ) t | = 1 (cid:12)(cid:12)(cid:12) t ψ ( t ) ψ (cid:48) (0) (cid:12)(cid:12)(cid:12) is smooth in ( − ε (cid:48)(cid:48) , ε (cid:48)(cid:48) ) for some small ε (cid:48)(cid:48) since | ψ (cid:48) (0) | = 1 .Based on definition of g , we can write g ( t ) = | t | α h ( t ) for some smooth function h ( t ) near 0.So since h ( t ) f ( t ) α is smooth and positive near 0 in Γ , we can extend it smoothlyto a neighborhood of x ∈ Σ . We denote this extension function as ˜ h ( z ) . Thenwe define ˜ g = | z | α ˜ h near x . This is a local extension of g near x which is positiveexcept at the point x since on ∂ Σ , we have ˜ g ( γ ( t )) = (cid:12)(cid:12) ϕ − ◦ γ ( t ) (cid:12)(cid:12) α (cid:18) | t || ϕ − ◦ γ ( t ) | (cid:19) α h ( t ) = h ( t ) | t | α . So by partition of unity, we can get a extension of ˜ g as we want. Moreover, we cankeep ˜ g positive on Σ \ ∂ Σ . Another part is essentially similar to this case. (cid:3)
So for our multiple junction surface M = ( θ Σ , · · · , θ q Σ q ; Γ) , the norm of secondfundamental form | A i | will belong to ˜ C + (Σ i ) if Σ i is non-flat. This is because when Σ i is minimal, Gauss map ν i ( x ) : Σ i → S will be the holomorphic map. So | A i | = | dν i | . Hence | A i | will have form | z | k f ( z ) near each zeros of | A i | with f ( z ) positive and smooth for some k ∈ Z + in some conformal coordinate. Hence, | τ (log | A i | ) | A i | p − | ≤ C | z | p − + C | z | p − where C i only depends on f . Since (cid:82) ε − ε | t | p − dt, (cid:82) ε − ε | t | p − are all finite, we know (cid:90) ( p − | τ (log | A i | ) | | A i | p − | φ i | p dµ Γ is locally integrable for each φ i ∈ L ∞ loc (Γ) and p > .Moreover, we also have | A | α ∈ L ∞ loc ( M ) ∩ H loc ( M ) by expand the gradient of | A i | near its zeros.5. L p estimate for the multiple junction surfaces In this section, we will prove the Theorem 1.4.For convenience, we use the following notation. For any φ ∈ H ( M ) with com-pact support, we write (cid:90) Σ φ := q (cid:88) i =1 (cid:90) Σ i φ i θ i dµ Σ i and (cid:90) Γ φ := q (cid:88) i =1 (cid:90) Γ φ i θ i dµ Γ for the integration on M = ( θ Σ , · · · , θ q Σ q ; Γ) .So the stability inequality can be write as (cid:90) Σ |∇ Σ φ | − | A | φ − (cid:90) Γ φ H Γ · τ ≥ for φ ∈ H ( M ) with compact support satisfying (2.2). Theorem 5.1.
Suppose M = ( θ Σ , · · · , θ q Σ q ; Γ) is a minimal multiple junctionsurface. Assume M is stable and complete. Let φ ∈ H ( M ) ∩ L ∞ ( M ) such thatsign ( φ ) | A | p − | φ | p satisfies compatible condition (2.2). Then (cid:90) Σ | A | p | φ | p ≤ C (cid:90) Σ | A | p − | φ | p − |∇ Σ φ | + (cid:90) Γ (cid:20) p − | τ (log | A | ) | − H Γ · τ (cid:21) | A | p − | φ | p . (5.1) Moreover, if φ ∈ W , p ( M ) ∩ L ∞ ( M ) , we also have (cid:90) Σ | A | p | φ | p ≤ C (cid:48) (cid:90) Σ |∇ Σ φ | p + C (cid:48) (cid:90) Γ [( p − | τ (log | A | ) | − H Γ · τ ] | A | p − | φ | p . (5.2) Here, we assume < p < , and C , C (cid:48) , C (cid:48) will only depend on M , They do notdepend on p . Note that if the Σ i is flat, we can define (cid:90) Γ | τ i (log | A i | ) | | A i | p − | φ i | p = 0 So the right hand side of (5.1) will always well-defined as we want.
URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY11
Remark.
Although the L p estimate (5.2) is the one appearing the original paper[1], we still need the slightly stronger version one like (5.1) in the later applicationsince the condition φ ∈ W , p ( M ) ∩ L ∞ ( M ) is not always satisfied based our choiceof functions. Proof of Theorem 5.1.
Let first consider the case that every Σ i is non-flat. Thismeans every | A i | has only isolated zeros on Σ i .We suppose φ ∈ L ∞ ( M ) ∩ H ( M ) with compact support. So by Hölder’s in-equality, we know | A | p − φ ∈ L ∞ ( M ) ∩ H ( M ) . Replacing φ by | A | p − φ in thestability inequality, we have(5.3) (cid:90) Σ | A | p φ ≤ ( p − (cid:90) Σ | A | p − |∇ Σ | A || φ + (cid:90) Σ | A | p − |∇ Σ φ | + 2( p − (cid:90) Σ | A | p − φ ∇ Σ | A | · ∇ Σ φ − (cid:90) Γ | A | p − φ H Γ · τ. Note that the compatible condition for φ is the condition that | A | p − φ satisfies(2.2).On the minimal surface Σ i , we have the Simon’s identity (see [7] for example),we have | A i | ∆ Σ i | A i | + | A i | = |∇ Σ i | A i || where ∆ Σ i is the Laplacian operator on Σ i .Multiplying | A i | p − φ i to the both side of Simon’s identity and integrating bypart, we have(5.4) (cid:90) Σ | A | p − |∇ Σ | A || φ = (cid:90) Σ | A | p φ − (2 p − (cid:90) Σ | A | p − |∇ Σ | A || φ − (cid:90) Σ | A | p − φ ∇ Σ φ · ∇ Σ | A | + (cid:90) Γ | A | p − τ ( | A | ) φ . Note that all the terms are finite in above identity by the properties of | A i | .Moving the second term in the right hand side of the above identity, we have p − (cid:90) Σ | A | p − |∇ Σ | A || φ = (cid:90) Σ | A | p φ − (cid:90) Σ | A | p − φ ∇ Σ φ · ∇ Σ | A | + (cid:90) Γ | A | p − τ ( | A | ) φ . (5.5) Substituting (5.5) in (5.3) and using Cauchy inequality, we get (cid:90) Σ | A | p φ ≤ p − (cid:90) Σ | A | p φ − ( p − (cid:90) Σ | A | p − φ ∇ Σ φ · ∇ Σ | A | + p − (cid:90) Γ | A | p − τ ( | A | ) φ + (cid:90) Σ | A | p − |∇ Σ φ | + 2( p − (cid:90) Σ | A | p − φ ∇ Σ φ · ∇ Σ | A | − (cid:90) Γ φ | A | p − H Γ · τ = p − (cid:90) Σ | A | p φ + ( p − (cid:90) Σ | A | p − φ ∇ Σ φ · ∇ Σ | A | + (cid:90) Σ | A | p − |∇ Σ φ | + p − (cid:90) Γ | A | p − τ (log( | A | )) φ − (cid:90) Γ φ | A | p − H Γ · τ ≤ ( p − (cid:90) Σ | A | p φ + (cid:18) p − (cid:19) (cid:90) Σ | A | p − |∇ Σ φ | + p − (cid:90) Γ | A | p − φ τ (log( | A | )) − (cid:90) Γ | A | p − φ H Γ · τ. (5.6)So for < p < , we have (cid:90) Σ | A | p φ ≤ (cid:90) Σ | A | p − |∇ Σ φ | + p − (cid:90) Γ | A | p − φ τ (log( | A | )) − (cid:90) Γ | A | p − φ H Γ · τ. (5.7)This is exactly the first inequality (5.1) we want to proof.Now let’s change to the case φ ∈ W ,p ( M ) ∩ L ∞ ( M ) . Recall the Young’s in-equality that for any x, y > , a, b > with a + b = 1 , we have xy ≤ x a a + y b b . We choose a = pp − , b = p , then we have | A | p − | φ | p − |∇ Σ φ | ≤ p − p | A | p | φ | p + 1 p |∇ Σ φ | p . So after replacing φ by sign ( φ ) | φ | p in (5.3), we have (cid:18) − p − p (cid:19) (cid:90) Σ | A | p | φ | p ≤ p (cid:90) Σ |∇ Σ φ | p + p − (cid:90) Γ | A | p − | φ | p τ (log( | A | )) − (cid:90) Γ | A | p − | φ | p H Γ · τ. (5.8)This time, we’ve assumed φ ∈ W ,p ( M ) ∩ L ∞ ( M ) with compact support andthe compatible condition for φ is sign ( φ ) | A | p − | φ | p will satisfy (2.2). Note thatsign ( φ ) | φ | p ∈ W , ( M ) ∩ L ∞ ( M ) , so the replacement is valid. URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY13
So for < p < , we have (cid:90) Σ | A | p | φ | p ≤ (cid:90) Σ |∇ Σ φ | p + 2( p − (cid:90) Γ | A | p − | φ | p τ (log( | A | )) − (cid:90) Γ | A | p − | φ | p H Γ · τ ≤ (cid:90) Σ |∇ Σ φ | p + 4 (cid:90) Γ [( p − | τ (log( | A | )) | − H Γ · τ ] | A | p − | φ | p . (5.9)This is exactly what we want.If it happens that several minimal surfaces in { Σ , · · · , Σ q } are flat, and weassume | A i | cannot equal to 0 on the support of φ i on Σ i which is not flat, thenafter replacing φ by | A | p − φ , | A i | p − φ i will vanish on Σ i which is flat. So all theintegration will still make sense if we just drop the terms integrated on Σ i which isflat and all the formulas above are valid.So this estimate still holds for the general case. (cid:3) Proof of main theorem
In this section, we will choose a suitable test function to get our main theorem.Before that, let’s discuss the angle condition of Σ i along Γ first.Note that since ∂ Σ i is smooth in Σ i , the conormal vector fields τ i is a smoothnormal vector field along Γ . We say M = ( θ Σ , · · · , θ q Σ q ; Γ) has equilibrium angles along Γ if for all ≤ i, j ≤ q , the angles between τ i , τ j are constants along Γ .So if M has equilibrium angles along Γ , we can choose a smooth normal vectorfields W along Γ such that W has the constant angle with τ i and unit length along Γ . After choosing a orientation on the normal vector field, we can write the anglebetween τ i , W as α i , which is a constant function on Γ . By disturbing W a bit ifnecessary, we can assume α i ∈ (0 , π ) ∪ ( π , π ) ∪ ( π, π ) ∪ ( π , π ) for all ≤ i ≤ q .We denote this normal vector field as W and we will use it to construct our testfunction. Note that we have W · ν i = sin α i under some suitable orientation of Σ i ,which is non-zero and constant on Γ . By our choice of α i , we know that cos α i isnon-zero for each i . Theorem 6.1.
Let M = ( θ Σ i , · · · , θ q Σ q ; Γ) be a minimal multiple junction surfacein R . We assume M is complete, stable and has quadratic area growth. Further-more, we assume Γ is compact and has equilibrium angles along Γ . Then each Σ i is flat.Proof. First, let’s define a smooth cutoff function η ( t ) on R by η ( t ) = (cid:40) , t ≤ , t ≥ such that η ( t ) is a monotonically decreasing on R and | η (cid:48) ( t ) | < . First Case: None of Σ i is flat .Let’s write the L p estimate (5.1) with following notation (cid:90) Σ | A | p | φ | p ≤ C I + II − III where I = (cid:90) Σ | A | p − | φ | p |∇ Σ φ | II = (cid:90) Γ p − | τ (log | A | ) | | A | p − | φ | p III = (cid:90) Γ H Γ · τ | A | p − | φ | p (6.1)We will estimate these three terms one by one after choosing a suitable test function.We use T r (Γ) to denote the tubular neighborhood of Γ , i.e. we define T r (Γ) := { x ∈ M : d Γ ( x ) < r } . For simplicity, we assume | A | has no zeros in T (Γ) \ Γ . Otherwise, we can do arescaling of M if necessary.Define the cutoff function ρ r on M by ρ r ( x ) = η (cid:18) d Γ ( x ) r (cid:19) . So ρ r will has support in T r (Γ) and equal to 1 in T r (Γ) and |∇ Σ ρ r | < r . Wewill write ρ ( x ) := ρ ( x ) . Define c i = W · ν i = sin α i .Now let’s define a function g i on Γ by g i ( x ) = (cid:89) j =1 , ··· ,q,j (cid:54) = i | A j | . Since | A i | | Γ ∈ ˜ C + (Γ) , g i ( x ) ∈ ˜ C + (Γ) , we can extend g i ( x ) to the whole Σ i suchthat g i ∈ ˜ C + (Σ i ) and positive on Σ i \ ∂ Σ i by Lemma 4.3.Then we choose our φ on M as φ i = sign ( c i ) | c i | p (cid:18) ρg p − p i + ρ r − ρ (cid:19) for some r > .At first, we note that φ will satisfies the compatible condition (2.2) since on Γ , φ i = sign ( c i ) | c i | p g p − p i , andsign ( φ i ) | A i | p − | φ i | p = c i q (cid:89) i =1 | A i | p − = (cid:32) q (cid:89) i =1 | A i | p − (cid:33) W · ν i . Note that (cid:81) qi =1 | A i | ∈ L ∞ loc (Γ) by Trace Theorem (or just by the properties offunctions in ˜ C + (Γ) ).Now let’s check φ ∈ H ( M ) . Note that g p − p i is either smooth or has form f ( z ) | z | k ( p − p near an arbitrary point in Σ i for some smooth function f in con-formal coordinate, and | z | k ( p − p ∈ H loc ( M ) , so g p − p i ∈ H loc ( M ) . Note that ρ, ρ r ∈ W , ∞ ( M ) since they are Lipschitz functions with compact support, weget φ ∈ H ( M ) by Hölder’s inequality.So by Theorem 5.1, we can put our φ in the estimate (5.1). The goal of thefollowing proof is to make the terms I, II and III small enough with relatively large r by choosing p very close to and some suitable φ .Now let’s fix some ε > and some r > from now on. Estimation of III.
URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY15
Right now we do not know the sign of III. But if III < , we can rotate W by90 degrees in normal bundle to get a new vector field ˜ W along Γ . So ˜ c i := ˜ W · ν i .Then we can define the new test function ˜ φ ˜ φ i = sign ( ˜ c i ) | ˜ c i | p (cid:18) ρg p − p i + ρ r − ρ (cid:19) . Along Γ , we have | φ i | = | c i | p g p − p i , | ˜ φ i | = | ˜ c i | p g p − p i . If we define g = (cid:81) qi =1 | A i | along Γ , we have q (cid:88) i =1 H Γ · τ i θ i | A i | p − | ˜ φ i | p = q (cid:88) i =1 H Γ · τ i θ i ˜ c i g p − = g p − q (cid:88) i =1 H Γ · τ i θ i (cid:16) ˜ W · ν i (cid:17) = g p − q (cid:88) i =1 H Γ · τ i θ i ( W · τ i ) = g p − q (cid:88) i =1 H Γ · τ i θ i (cid:104) − ( W · ν i ) (cid:105) = g p − q (cid:88) i =1 − H Γ · τ i θ i ( W · ν i ) = − q (cid:88) i =1 H Γ · τ i θ i | A i | p − | φ i | p . Here we’ve used the fact that M is minimal along Γ . Hence ˜ III := (cid:90) Γ H Γ · τ | A | p − | ˜ φ | p = − III . So after replacing W by ˜ W , we can get(6.2) III > . Estimation of II.
Note that Γ is compact, | A | p − and | φ | p are all bounded on Γ uniformly withrespect to p ∈ (1 , ) . Since the integration (cid:82) Γ | τ (log | A | ) | is finite by the propertyof smooth finite order functions, the integration (cid:90) Γ | τ (log | A | ) | | A | p − | φ | p < ∞ . Hence we can choose p > very close to 1 such that for every p ∈ (1 , p ) , wehave(6.3) II < ε. Estimation of I.
The trick part for estimating I are the points that φ fails to be in W ,p ( M ) nearby, which are the zeros of φ . Denote P i = { x ∈ Σ i : g i ( x ) = 0 } . So by the definition of g i , we know P i ⊂ ∂ Σ i . Choose x ∈ P i , let’s consider the integrationI x,δ := (cid:90) Σ i ∩ B Mδ ( x ) | A i | p − | φ i | p − |∇ Σ i φ i | for δ < .In Σ i ∩ B Mδ ( x ) , we have | φ i | = | c i | p g p − p i . Still we work at conformal coordinatenear x and we choose δ small enough to make sure Σ i ∩ B Mδ ( x ) is in this coordinatechart. Then g i has form g i ( z ) = f ( z ) | z | l for some f ( z ) positive and smooth near x and l ∈ Z + .We compute | c i | − p |∇ Σ i φ i | = (cid:18) p − p (cid:19) g − p i |∇ Σ i g i | = (cid:18) p − p (cid:19) f − p | z | − lp (cid:12)(cid:12)(cid:12) | z | l ∇ Σ i f + lf | z | l − ∇ Σ i | z | (cid:12)(cid:12)(cid:12) ≤ (cid:18) p − p (cid:19) f − p (cid:16) | z | l − lp |∇ Σ i f | + l f | z | l − − lp |∇ Σ i | z || (cid:17) . (6.4)Note that |∇ Σ | z || might not equal to 1. Nerveless, it is bounded near x .Since f is smooth and positive so it has lower bound near x , we know the term f − p | z | l − lp |∇ Σ i f | is bounded near x . Hence the integration on this term can bearbitrary small by choose δ small enough.For the term l f − p | z | l − − lp |∇ Σ i | z || , if we assume the metric in this coordi-nate is λ ( z ) dzdz and l f − p |∇ Σ i | z || λ is bounded by C , then (cid:90) Σ i ∩ B Mδ ( x ) l f − p | z | l − − lp |∇ Σ i | z || ≤ (cid:90) | z | <δ and z ∈ Σ i C | z | l − − lp √− dz ∧ dz ≤ πC l − lp δ l − lp = πCpl ( p − δ l − lp . (6.5)Combining (6.4),(6.5), and noting | A i | p − | φ i | p − is bounded near x , we getI x,δ x ≤ C (cid:18) p − p (cid:19) (cid:18) ε (cid:48) + πCpl ( p − δ l − lp (cid:19) ≤ C (cid:18) p − p (cid:19) ε (cid:48) for some δ x small enough.Here, the constance C does not depend on p . So we can choose p x ∈ (1 , p ) smallenough such that if p ∈ (1 , p x ) , then I x,δ x < ε (cid:48) for any given ε (cid:48) > . Note that the set P i is a finite set for each ≤ i ≤ q , so wecan choose p = min x ∈∪ P i p x and δ = min x ∈∪ P i δ x . Denote B ∈ M as B := ∪ qi =1 ∪ x ∈ P i (Σ i ∩ B δ ( x )) . Then (cid:90) B | A | p − | φ | p − |∇ Σ φ | ≤ q (cid:88) i =1 θ i (cid:93) ( P i ) ε (cid:48) where (cid:93) ( P i ) denote the cardinality of the set P i . By requiring ε (cid:48) small, we can find p ∈ (1 , p ) , δ > such that(6.6) I (cid:48) = (cid:90) B | A | p − | φ | p − |∇ Σ φ | ≤ εC . URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY17
Now we focus on the estimate ofI (cid:48)(cid:48) = (cid:90) Σ \ B | A | p − | φ | p − |∇ Σ φ | . Clearly we have(6.7) I = I (cid:48) + I (cid:48)(cid:48) . Note that right now we know φ is positive and smooth with compact support in Σ \ B , so φ ∈ W , ∞ ( M ) . So we can apply Young’s inequality to get(6.8) I (cid:48)(cid:48) ≤ p − p (cid:90) Σ \ B | A | p | φ | p + 1 p (cid:90) Σ \ B |∇ Σ φ | p < ∞ . So we choose p ∈ (1 , p ) small such that(6.9) p − p < C so that the first term in above inequality can be absorbed by left hand side of (5.1).For the second term in (6.8), we have (cid:90) Σ \ B |∇ Σ φ | p = (cid:90) T (Γ) \ B |∇ Σ φ | p + (cid:90) T r (Γ) \ T r (Γ) |∇ Σ φ | p =: I + I . (6.10)In T (Γ) \ B , we know | φ i | = | c i | p ρ (cid:18) g p − p i − (cid:19) + | c i | p . Hence |∇ Σ i φ i | p = | c i | (cid:12)(cid:12)(cid:12)(cid:12) p − p g − p i ∇ Σ i g i ρ + ∇ Σ i ρ (cid:18) g p − p i − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ p c i (cid:18) p − p (cid:19) p g − i ρ p |∇ Σ i g i | p + 2 p c i |∇ Σ i ρ | p (cid:12)(cid:12)(cid:12)(cid:12) g p − p i − (cid:12)(cid:12)(cid:12)(cid:12) p ≤ C ( p − g − i |∇ Σ i g i | p + C (cid:12)(cid:12)(cid:12)(cid:12) g p − p i − (cid:12)(cid:12)(cid:12)(cid:12) p . (6.11)Note that g i has positive upper and lower bound on T (Γ) \ B , |∇ Σ i g i | has anupper bound on T (Γ) \ B since it is smooth. Hence, as p → + , g i → uniformly.So we can choose p ∈ (1 , p ) small enough to make(6.12) I < εC for any p ∈ (1 , p ) . From now on, we will fix p ∈ (1 , p ) .For the integration I , we haveI = (cid:90) T r (Γ) \ T r (Γ) c |∇ Σ ρ r | p ≤ C (cid:90) T r (Γ) \ T r (Γ) r p ≤ Cr − p By area growth condition . (6.13)Here the constant C does not depend on p and r . Since we’ve fixed p , we canchoose r > r so large, such that(6.14) I ≤ Cr − p < εC . Now, let’s combine the estimate (6.14), (6.12), (6.10), (6.8), (6.9), (6.7), (6.3),(6.2), (6.1) with (5.1) to get(6.15) (cid:90) Σ | A | p | φ | p ≤ (cid:90) Σ \ B | A | p | φ | p + ε + ε + ε + ε. So(6.16) (cid:90) Σ | A | p | φ | p ≤ ε. By definition of φ , we have (cid:90) T r (Γ) \ T (Γ) min { , | A | } c ≤ (cid:90) T r (Γ) \ T (Γ) | A | p c ≤ (cid:90) Σ | A | p | φ | p ≤ ε. (6.17)So the left hand side of the above inequality does not depend on r and p . Byarbitrary choice of ε and noting c i (cid:54) = 0 on each Σ i , we know actually | A | = 0 on T r (Γ) \ T (Γ) , and thus this implies each Σ i is flat, which contradicts our assump-tion that each Σ i is non-flat. Second case: One of Σ i is flat. For the case that one of Σ i being flat, we suppose Σ is lying the plane P .Clearly, if there are another Σ i , which is flat and different from Σ , Σ i shouldbe lying P , too since we’ve assume Γ is compact. Again, we write it as Σ forsimplicity. So the only possible choice of Σ is Σ = P \ (Σ ∪ ∂ Σ ) . Note that P is stable, so we can remove it from this triple junction surface to get the remainingone (( θ − θ )Σ , θ Σ , · · · , θ q Σ q ; Γ) or (( θ − θ )Σ , θ Σ , · · · , θ q Σ q ; Γ) is unstable.So WOLG, we assume there are only one Σ i , which we call it Σ , is flat.This time we choose W = τ . The trick park is the term III might not have afavorable sign. So we need a bit more precise estimation of the total curvature.Let’s use K i to denote the sectional curvature on Σ i . Following from B. White’sproof ([8]), we have the following lemma. Lemma 6.2.
For each Σ i which is non-compact with boundary Γ , we have (cid:90) Γ − H Γ · τ i ≤ (cid:90) Σ i − K Σ i where the K Σ i is the sectional curvature of Σ i .Proof. Fix a point p ∈ Γ , and we define B r := { p ∈ Σ i : d i ( p, p ) < r } . We write Γ r = ∂B r \ Γ , the remaining boundary part of B r except Γ .Note that we can choose a large r such that Γ ⊂ B r . So for r > r , we know Γ and Γ r do not connect with each other.By the result of P. Hartman [9], we know Γ r is, for almost all r , a piecewisesmooth, embedded closed curve in Σ i . So we can apply Gauss-Bonnet theorem toget (cid:90) B r K Σ i + (cid:90) Γ H Γ · ( − τ i ) + (cid:90) Γ r κ g + (cid:88) ( exterior angles of Γ r )= 2 πχ ( B r ) = 2 π (2 − h ( r ) − c ( r )) (6.18) URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY19 where h ( r ) and c ( r ) are the number of handles and the number of boundary compo-nents, respectively, of B r with r > r . Here we also use κ g to denote the curvatureof Γ r inside the surface Σ i with respect to the inner conormal vector field.Let L ( r ) be the length of Γ r . So by the first variation formula of piecewisesmooth curve, we have L (cid:48) ( r ) = (cid:90) Γ r κ g + (cid:88) exterior angles of Γ r . We also note that c ( r ) ≥ since Γ has at least one component and Γ r is alwaysnon-empty since we’ve assume Σ i is complete and not compact for any r > r . Socombining with the above Gauss-Bonnet formula we’ve got, we can get − (cid:90) Γ H Γ · τ i + L (cid:48) ( r ) ≤ − (cid:90) B r K Σ i . Note that L ( r ) > for all r > r , we have lim sup r →∞ L (cid:48) ( r ) ≥ . And since K Σ i ≤ , we can take r → ∞ to get − (cid:90) Γ H Γ · τ i ≤ − (cid:90) Σ i K Σ i . This is what we want. (cid:3)
Let’s go back to the proof of main theorem. Again, we choose the function g i on Γ as g i ( x ) = (cid:89) j =2 , ··· ,q,j (cid:54) = i | A j | and choose our φ on M as φ := φ r,p = sign ( c ) | c | p (cid:16) ρg p − p + ρ r − ρ (cid:17) where c i = W · ν i . Here we use subscript to indicate φ depends on r and p ifneeded. Clearly, sign ( φ i ) | A i | p − | φ i | p satisfies the compatible condition.This time, we do not have a good sign for the term III, so we keep it in ourestimate. Based on essentially same argument, we can get a similar estimate like(6.15) as(6.19) (cid:90) Σ | A | p | φ | p ≤ (cid:90) Σ \ B | A | p | φ | p + 4 ε − IIIfor p small enough and r large enough which might depend on p . If it happens thatIII ≥ , the previous argument shows that each Σ i for i = 2 , · · · , q is flat.For simplicity, we write III in the form which depends on p asIII p := (cid:90) Γ | A | p − | φ | p H Γ · τ = (cid:90) Γ g p − c H Γ · τ where g = (cid:81) qi =2 | A i | . So if we write III , we just meansIII := (cid:90) Γ c H Γ · τ. We note g p − → a.e. on Γ since the zeros of g are isolated, so by DominatedConvergence theorem, we have (cid:90) Γ g p − c H Γ · τ → (cid:90) Γ c H Γ · τ as p → . This means we can choose p large to make(6.20) | III p − III | ≤ | III | since III < as we’ve assumed. Similarly, g p − p i → a.e. on Σ i and | A i | p → | A i | a.e. on Σ i , we have (cid:90) Σ | A | p | φ | p → (cid:90) Σ | A | c ρ r as p → .Hence, for some fixed r , we can always choose p small enough to make sure(6.21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Σ | A | p | φ | p − | A | c ρ r (cid:12)(cid:12)(cid:12)(cid:12) ≤ | III | . By our previous lemma 6.2, we have (cid:90) Σ | A | c = − (cid:90) Σ K Σ c ≥ − (cid:90) Γ c H Γ · τ = 2 | III | . Note (cid:82) Σ | A | c ρ r → (cid:82) Σ | A | c , so we can fix a r large enough to get(6.22) (cid:90) Σ | A | c ρ r ≥ | III | . Combining with (6.21), we have (cid:90) Σ | A | p | φ r ,p | p ≥ | III | for p sufficient small. Note φ r,p is an increasing function with respect to variable r as r > , so we actually have(6.23) (cid:90) Σ | A | p | φ r,p | p ≥ | III | for all r > r , < p < p for some p sufficient closed to 1.Hence, we can choose p small enough again, and then choose r > r large enoughto make the estimate (6.19) holds for ε = | III | . So (6.19) implies (cid:90) Σ | A | p | φ | p ≤ | III | − III p ≤ | III | + 32 | III | = 53 | III | where we’ve used (6.20). This is a contradiction with the estimate (6.23). So it isimpossible that only Σ is flat. Hence, we finished our proof. (cid:3) Theorem 6.3.
Let M = ( θ , Σ , · · · , θ q Σ q ; Γ) be a minimal multiple junction sur-face in R . We assume M is complete, stable and has quadratic area growth. Fur-thermore, we assume Γ is a straight line and M has equilibrium angles along Γ ,then each Σ i is flat.Proof. This case is much easier than the case of Γ compact. Note that for any i (cid:54) = j , by rotating Σ i along Γ for a suitable angle, we can make Σ i and Σ j share thesame outward conormal of boundary. Hence by Hopf’s boundary lemma, Σ i willbe identical to Σ j after rotation.This says that each pieces of surface in M are all isometric to each other. More-over, by reflection principle, we also have τ i ( | A i | ) = 0 for each ≤ i ≤ q . Hencethe stability operator for M is (cid:90) Σ | A | φ ≤ (cid:90) Σ |∇ Σ φ | for some φ = W · ν . To apply the L p estimate, we need sign ( φ ) | A | p − | φ | p tosatisfy the compatible condition (2.2). Note that | A i | = | A j | , we only need torequire sign ( φ ) | φ | p to satisfy (2.2).Now, as usual we choose W having constant angles with each τ i and make sure c i := W · ν i (cid:54) = 0 for each i . Now we choose an arbitrary point x on Γ . Define cutoff URVATURE ESTIMATES FOR STABLE MINIMAL SURFACES WITH A COMMON FREE BOUNDARY21 function ρ r ( x ) support in B M r ( x ) which equals to 1 in B Mr ( x ) and has gradient lessthan r .So we can choose our φ as φ = sign ( c ) | c | p ρ r . Hence we can apply L p estimate to get (cid:90) Σ | A | p | φ | p ≤ C (cid:90) Σ |∇ Σ φ | p . Standard argument in [1] will imply each Σ i will be flat. (cid:3) Remark.
Indeed, we can remove the quadratic area growth condition in the case Γ a straight line. Actually, one can just use the result of Bernstein theorem forstable minimal surface to get this result.As a corollary, we can also get some result related to the stable capillary minimalsurface. Corollary 6.4.
Let P be a plane in R . Then there is no (oriented) stable completeminimal surface Σ with boundary ∂ Σ such that ∂ Σ ∈ P , ∂ Σ compact, and Σ hasconstant angle with P along ∂ Σ . Here the stability of the capillary minimal surface means this surface is stable ofcapillary energy under the variation fixing the plane P .This result is an immediately result in the proof of Theorem 6.1. The variationwe’ve taken in the proof of second case is just the one fixing the plane P .7. Further questions
As we’ve seen, we still leave some questions related to minimal multiple junctionsurface.The first one is, what if Γ is neither compact nor straight line? Question 7.1.
Can we relax the condition of Γ to be compact and straight line, sothat we still get the similar Bernstein theorem for stable minimal multiple junctionsurface? In particular, we want to know the stability property of universal cover of Y -shaped bent helicoid.Another question is, we still need the help of area growth to get the control ofcurvature. So we may still want to remove this condition in some sense. Question 7.2.
Can one get the quadratic area growth for stable minimal multiplejunction surfaces like the result in [2] in some sense?
In [2], one may need simply connected condition to get the quadratic area growth.This is not a problem when taking about usual smooth surface since we can alwaystake a universal cover without affecting stability. But things get unusual especiallywhen requiring Γ compact. Moreover, one may still need a careful considerationwhen taking about the simply connected of multiple junction surface.The last question is related to higher dimension case. Question 7.3.
Can one get the similar Bernstein theorem for stable minimal mul-tiple junction hypersurface with the hypersurface dimension greater than 2?
As we’ve seen, basically we can still get the similar L p estimation of curvaturelike Schoen, Simon, Yau’s result [1] for ≤ n ≤ . But we have an additionalcompatible condition along Γ , so we need p → in order to get the curvatureestimation near Γ . This will force our surface dimension to be . References [1] Richard Schoen, Leon Simon, and Shing-Tung Yau. Curvature estimates for minimal hyper-surfaces.
Acta Mathematica , 134(1):275–288, 1975.[2] Tobias H Colding and William P Minicozzi. Estimates for parametric elliptic integrands.
In-ternational Mathematics Research Notices , 2002(6):291–297, 2002.[3] William H Meeks and Matthias Weber. Bending the helicoid.
Mathematische Annalen ,339(4):783–798, 2007.[4] Brian Krummel. Regularity of minimal hypersurfaces with a common free boundary.
Calculusof Variations and Partial Differential Equations , 51(3-4):525–537, 2014.[5] Leon Simon et al. Cylindrical tangent cones and the singular set of minimal submanifolds.
Journal of Differential Geometry , 38(3):585–652, 1993.[6] Harold Rosenberg. Hypersurfaces of constant curvature in space forms.
Bull. Sci. Math ,117(2):211–239, 1993.[7] Tobias H Colding and William P Minicozzi.
A course in minimal surfaces , volume 121. Amer-ican Mathematical Soc., 2011.[8] Brian White et al. Complete surfaces of finite total curvature.
Journal of Differential Geometry ,26(2):315–326, 1987.[9] Philip Hartman. Geodesic parallel coordinates in the large.
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Department of Mathematics, The Chinese University of Hong Kong, Shatin, HongKong
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