A new proof of a characterization of small spherical caps
aa r X i v : . [ m a t h . DG ] J un A new proof of a characterizationof small spherical caps
Rafael L´opez ∗ Departamento de Geometr´ıa y Topolog´ıaUniversidad de Granada18071 GranadaSpainemail: [email protected]
Abstract
It is known that planar disks and small spherical caps are the only constantmean curvature graphs whose boundary is a round circle. Usually, the proofinvokes the Maximum Principle for elliptic equations. This paper presentsa new proof of this result motivated by an article due to Reilly. Our proofutilizes a flux formula for surfaces with constant mean curvature together withintegral equalities on the surface.
A surface in Euclidean space R with the property that its mean curvature is constantat each point is called a constant mean curvature surface or CMC surface for short.Round spheres are closed CMC surfaces. Here by closed surface we mean compactand without boundary surface. A famous theorem due to Hopf asserts that anyclosed CMC surface of genus 0 must be a round sphere [4]. Later, Alexandrov provedin 1956 that any embedded closed CMC surface in R must be a round sphere [1].For a long time it was an open question whether or not spheres were the only closedCMC surfaces in R . If a such surface were to exist, it would necessarily be a surface ∗ Partially supported by MEC-FEDER grant no. MTM2007-61775. C is a circle of radius r >
0, we consider C in a sphere S ( R ) of radius R , R ≥ r . The mean curvature of S ( R ) is H = 1 /R with respect to the inward orientation. Then C splits S ( R ) in two spherical capswith the same boundary C and constant mean curvature H . If R = r , both capsare hemispheres whereas if R > r , there are two geometrically distinct caps whichwe call the small and the big spherical cap. On the other hand, the planar diskbounded by C is a compact surface with constant mean curvature H = 0. Thesesurfaces are the only totally umbilic compact CMC surfaces bounded by C .In 1991, Kapouleas found other examples of CMC surfaces bounded by a circle [5].The surfaces that he obtained have higher genus and self-intersections. Thus, oneasks under what conditions a compact CMC surface bounded by a circle is spherical.Taking into account the theorems of Hopf and Alexandrov for closed surfaces abovecited, the natural hypotheses to consider for surfaces bounded by a circle is thateither S has the simplest possible topology, that is, the topology of a disk, or that S is embedded. Surprisingly, we haveConjeture 1. Planar disks and spherical caps are the only compact CMCsurfaces bounded by a circle that are topological disks .Conjeture 2. Planar disks and spherical caps are the only compact CMCsurfaces bounded by a circle that are embedded .This means that our knowledge about the structure of the space of CMC surfacesbounded by a circle is quite limited and only several partial results have been ob-tained by different authors (we refer to [6] again). Of course, the methods of prooffor the Hopf and Alexandrov Theorems can not be applied with complete successin the context of a non-empty boundary. This fact, together the lack of examples,suggests that although the problems in the non-empty boundary case have the sameflavor as in the closed one, the proofs are more difficult.A partial answer to the conjecture 2 is the following Theorem 1 (Alexandrov)
Let C be a round circle in a plane Π and let S be anembedded compact CMC surface bounded by C . If S lies in one side of Π , then S isa planar disk or a spherical cap. S lies on one side of the plane containingthe boundary . Although Alexandrov did not state this result, the proof is accom-plished using the same technique that he used in proving his theorem that wasstated above: the so-called Alexandrov reflection method. Behind this method liesthe classical Maximum Principle for elliptic partial differential equations, togetherwith the moving plane technique. A particular case of this Theorem is the following.Given H and a circle C , among the two spherical caps bounded by C with meancurvature H , only the small one is a graph. Using this method, we characterize thesmall spherical caps as Theorem 2
Let C be a round circle in a plane Π and let S be a compact CMCsurface bounded by C . If S is a graph over Π , then S is a planar disk or a smallspherical cap. The purpose of this article is to give a new proof of this result and that does notinvolve the Maximum Principle . This different approach, that is, avoiding the Max-imum Principle, appeared in the closed case which motivated the present work. In1978, Reilly obtained another proof of the Alexandrov theorem for CMC closed sur-faces without the use of the Maximum Principle thanks to a combination of theMinkowski formulae with some new elegant arguments [12]. In this sense, a newelementary proof of Alexandrov’s Theorem due to Ros appears in [13].In the same spirit, we will use integral formulae together a type of ”flux formula”.Moreover we will see in the next section how the equation that characterizes a CMCsurface can be expressed in terms of the Laplace operator. This was already noticedby Reilly as one can see from the title of his article. This fact allows us to establishsome geometrical properties about CMC surfaces using basic multivariable Calculus.
Let S be a surface in R and which we write locally as the graph of a smooth function f , z = f ( x, y ), ( x, y, z ) being the usual coordinates of R . We orient S with thechoice of normal given by N ( x, y, f ( x, y )) = ( − f x , − f y , q f x + f y ( x, y ) , (1)where the subscripts indicate the corresponding partial derivatives. The mean cur-vature H of S satisfies the following partial differential equation:2 H (1 + f x + f y ) = (1 + f y ) f xx − f x f y f xy + (1 + f x ) f yy . (2)3quation (2) may written asdiv ∇ f q |∇ f | = 2 H, (3)where div and ∇ stand for the divergence and gradient operators respectively. InPDE theory, this equation falls into the category of elliptic type, whose main prop-erty is the existence of a Maximum Principle: if two functions f and f satisfyequation (3) with the same Dirichlet condition, then f = f . We refer the readerto [3, sect. 10.5]. For CMC surfaces, this geometrically translates to the assertionthat if two CMC surfaces with the same constant mean curvature intersect tangen-tially at some point and one surfaces lies locally on one side of the other, then bothsurfaces must coincide in a neighborhood of that point. This property was used byAlexandrov in proving his theorem by using a surface as a comparison surface withitself in a reflection process. When the surface has non-empty boundary, we mustadd the extra hypothesis that S lies over Π as states Theorem 1. By doing this, weavoid the presence of a possible contact between an interior point with a boundarypoint of the surface where the Maximum Principle fails. We refer the reader to [6, 7]for detailed proofs of Theorem 1.We prove two results about CMC surfaces which have their own geometric interest(they will not be used later). Both results are well known in the literature althoughusually the Maximum Principle is invoked in the proofs. However, we show themby using a basic knowledge of calculus and differential geometry. Theorem 3
Let D be a domain of a plane Π and let S be a compact CMC graphon D whose boundary is ∂D . If H = 0 , then int ( S ) lies in one side of Π .Proof : We argue by contradiction. Assume that S has (interior) points on bothsides of Π. Consider p = ( x , y , z ), z > p = ( x , y , z ), z ≤
0, pointsof S with highest and lowest height z , respectively. If S is the graph of a function z = f ( x, y ), then, f x ( x i , y i ) = f y ( x i , y i ) = 0 , i = 0 , f xx ( x , y ) , f yy ( x , y ) ≤ , f xx ( x , y ) , f yy ( x , y ) ≥ . (5)Using the orientation given by (1), let us compute the mean curvature H at p and p . Because H is constant and using (4) and (5), equation (2) leads to2 H = 2 H ( p ) = ( f xx + f yy ) ( x , y ) ≤ ≤ ( f xx + f yy ) ( x , y ) = 2 H ( p ) = 2 H. (6)Since H = 0, we get a contradiction. q.e.d H ( p ) = ∆ f ( x , y ) ≤ ≤ ∆ f ( x , y ) = 2 H ( p ) , where ∆ = ∂ xx + ∂ yy is the Euclidean Laplacian. This indicates that under a certainchoice of coordinates, (2) can be expressed in terms of the Laplace operator. Seealso [12].We treat the minimal case, that is, H = 0. Theorem 4
Consider a Jordan curve C in a plane Π . If S is a compact CMCsurface with H ≡ , whose boundary is C , then S is the planar domain D thatbounds C .Proof : We point out that we have dropped the hypothesis that S is a graph. Weuse a similar proof as in Theorem 3 and we follow the notation used there. Thereasoning is by contradiction again. Without loss of generality, we assume that S has points over Π. Let Γ ⊂ Π be a circle of radius r sufficiently large so that D liesstrictly inside of the circular disk determined by Γ and so that the hemisphere K with ∂K = Γ over Π also lies over S . Let S ( H ) be the family of small spherical capsover Π with ∂S ( H ) = Γ and parameterized by their mean curvature H oriented by(1). Then − /r < H <
0. In the limit case, S ( − /r ) = K . Beginning from thevalue H = − /r , we let H → h , − /r < h <
0, that S ( h )touches the original surface S . Let p be the contact point. Both surfaces are locallygraphs of functions defined in the (common) tangent plane at p . This point is notnecessarily the highest point of S , but we do a change of coordinates so that p isthe origin, the tangent plane of S ( h ) and S at p is the xy -plane and S ( h ) lies over S in a neighborhood of p . Now p is the highest point of S and both surfaces liebelow Π.Consider the two functions f and g whose graphs are S and S ( h ) respectively anddefined in some planar domain Ω of Π containing the origin. Let u = f − g . Then u ≤ , g > ∂D , p is an interiorpoint of Ω. Consequently, f x (0 ,
0) = f y (0 ,
0) = g x (0 ,
0) = g y (0 ,
0) = 0 and u xx (0 , , u yy (0 , ≤ . (7)However at the point p , equation (2) for S and S ( h ) is0 = ( f xx + f yy )(0 ,
0) and − r = ( g xx + g yy )(0 , , /r = ( u xx + u yy )(0 , > q.e.d We have seen that if C is a circle of radius r , the possible values of the meancurvatures H for spherical caps bounded by C lies in the range [ − /r, /r ] because R ≥ r for the radius of the spheres S ( R ). Thus, the boundary C imposes restrictionson the possible values of mean curvature. We show that this occurs for a generalcurved boundary. Consider a compact CMC surface S with boundary ∂S = C andlet Y be a variation field in R . The first variation formula of the area | A | of thesurface S along Y is δ Y | A | = − H Z S h N, Y i d S − Z ∂S h ν, Y i d s, where N is the unit normal vector of S , H is the mean curvature relative to N , ν represents the inward unit vector along ∂S and d s is the arc-length element of ∂S . Let us fix a vector ~a ∈ R and consider Y as the generating field of a family oftranslations in the direction of ~a . As Y generates isometries of R , the first variationof A vanishes and thus 2 H Z S h N, ~a i d S + Z ∂S h ν, ~a i d s = 0 . (8)The first integral transforms into an integral over the boundary as follows. Thedivergence of the field Z p = ( p × ~a ) × N , p ∈ S , is − h N, ~a i (here × denotes thecross product of R ). The Divergence Theorem, together with (8), yields − H Z ∂S h α × α ′ , ~a i d s = Z ∂S h ν, ~a i d s, (9)where α is a parametrization of ∂S such that α ′ × ν = N .This equation known as the “balancing formula” or “flux formula” is due to R.Kusner (see [9]; also in [10, 8]). It is a conservation law in the sense of Noetherthat reflects the fact that the area (the potential) is invariant under the group oftranslations of Euclidean space. On the other hand, if D is a 2-cycle with boundary ∂S , the formula can be viewed as expressing the physical equilibrium between the6orce of exterior pressure acting on D (the left-hand side in (9)) with the force ofsurface tension of S that act along its boundary (the right-hand side).If the boundary C lies in the plane Π = { x ∈ R ; h x, ~a i = 0 } , for | ~a | = 1, then (9)gives 2 H ¯ A = Z ∂S h ν, ~a i d s, (10)where ¯ A is the algebraic area of C . Given a closed curve C ⊂ R that bounds adomain D , and noting that h ν, ~a i ≤
1, the possible values of the mean curvature H of S satisfy | H | ≤ length( C )2 area( D ) . (11)In particular, if C is a circle of radius r > , a necessary condition for the existenceof a surface spanning C with constant mean curvature H is that | H | ≤ /r . Remark 1 If S is the graph of z = f ( x, y ) it follows from the Divergence Theoremand (3) that | H | area( D ) = (cid:12)(cid:12)(cid:12)(cid:12)Z D H d D (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂D h ∇ f q |∇ f | , ~n i d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∂D |∇ f | q |∇ f | d s < Z ∂D s = length( C ) , where ~n is the unit normal vector to ∂D in Π . Then, | H | < length(C)2 area(D) . As consequence of Remark 1, the proof of Theorem 2 is very simple by using theMaximum Principle as we show at this time. If S is the graph of a function z = f ( x, y ) and H is its mean curvature, then | H | < /r , where r is the radius of C .But there exists a small spherical cap with the same boundary and mean curvatureas S . As this cap is the graph of a function f , we have that f and f are twosolutions of (3) with the same Dirichlet condition, the Maximum Principle implies f = f . Other proof of Theorem 2 using a combination of the flux formula and theMaximum Principle appears in [2]. 7 A new proof of Theorem 2
In this section we will prove our result without the use of the Maximum Principle.Let S satisfy the hypotheses of Theorem 2 and let ( x, y, z ) be the usual coordinatesof R . Without loss of generality, we assume that Π is the xy -plane, that is, Π = { z = 0 } and that C is a circle of radius r > ~a = (0 , , N given by (1). Then h N, ~a i > S . We willuse the notation of Section 3. First, let α be the parametrization of C such that α ′ × ν = N . We know that α ′′ = − α/r (independent of the orientation of C ) andsince h N, ~a i > ∂S , we have α × α ′ = r~a . Then h ν, ~a i = h N × α ′ , ~a i = h N, α ′ × ~a i = 1 r h N, α i . (12)The integral equation (9) gives − πr H = Z C h ν, ~a i d s. (13)and thus, equation (8) is Z S h N, ~a i d S = πr . (14)We will need the following result: Lemma 1
The function h N, ~a i satisfies ∆ h N, ~a i + | σ | h N, ~a i = 0 , (15) where ∆ is the Laplace-Beltrami operator on S and σ is the second fundamentalform.Proof : Formula (15) holds for any CMC surface. Let x : S → R be an immersionof a surface in R . For any vector field Y of the ambient space R , we consider thedecomposition Y = V + uN , where V is a tangent vector field to x and u = h N, Y i is the normal component of Y . We consider a smooth variation ( x t ) of x ( x = x )whose variation vector field is uN , that is, ∂ t ( x t ) t =0 = uN . Then the variation ofthe mean curvature H t of the ( x t ) changes according to ∂ t ( H t ) t =0 = 12 (∆ u + | σ | u ) + h∇ H, V i . x is a CMC surface. Then ∇ H = 0. Let us take the vector field Y = ~a whose associated one-parameter subgroup generates translations. Thus themean curvature is fixed pointwise throughout the variation, and so, ∂ t ( H t ) t =0 = 0,proving (15). q.e.d We follow with the proof of Theorem 2. By applying the Divergence Theorem tothe vector field ∇h N, ~a i and using equation (15), we get Z S | σ | h N, ~a i d S = Z C h dN ν, ~a i d s. (16)We study each side of (16) beginning with the left-hand side. The inequality | σ | ≥ H holds on any surface, and equality occurs only at umbilic points. By using (14)and that h N, ~a i > S , the left-hand side of (16) yields Z S | σ | h N, ~a i d S ≥ H Z S h N, ~a i d S = 2 πr H . (17)We now turn our attention to the right-hand side of (16). First, note dN ν = − σ ( α ′ , ν ) α ′ − σ ( ν, ν ) ν. From (12), we have σ ( ν, ν ) = 2 H − σ ( α ′ , α ′ ) = 2 H + h dN α ′ , α ′ i = 2 H − h N, α ′′ i = 2 H + 1 r h N, α i = 2 H + 1 r h ν, ~a i . (18)Because h α ′ , ~a i = 0, and using (13) and (18), we have Z C h dN ν, ~a i d s = − Z C σ ( ν, ν ) h ν, ~a i d s = − Z C (cid:16) H + 1 r h ν, ~a i (cid:17) h ν, ~a i d s = 4 πr H − r Z C h ν, ~a i d s. (19)We use (13) and the Cauchy-Schwarz inequality as follows: Z C h ν, ~a i d s ≥ πr Z C h ν, ~a i d s ! = 2 πr H . (20)9hen equation (19) and inequality (20) imply Z C h dN ν, ~a i d s ≤ πr H . (21)Finally, by combining (16), (17) and (21), we obtain2 πr H ≤ Z S | σ | h N, ~a i d S = Z C h dN ν, ~a i d s ≤ πr H . Therefore, we have equalities in all the above inequalities. In particular, | σ | = 2 H on S . This means that S is a totally umbilic surface of R and so, it is an open ofa plane or a sphere. Because, the boundary of S is a circle, S is a planar disk or ita spherical cap. In the latter case, S must be the small spherical cap since S is agraph. This concludes the proof of the theorem.We point out that the hypothesis that S is a graph has been only used in inequality(17) to assert | σ | h N, ~a i ≥ H h N, ~a i . However, the rest of the proof is valid forany CMC compact surface bounded by a round circle. The reader may then try touse the ideas that underlie the proof of Theorem 2 to derive other results. As anexample, we show the following theorem where we replace the hypothesis that thesurface is a graph by the hypothesis of non negativity of the Gauss curvature. Again,we follow the spirit of our work and we do not invoke the Maximum Principle. Theorem 5
Let S be a compact CMC surface bounded by a round circle C . If theGauss curvature K is non-negative, then S is a planar disk or a spherical cap.Proof : As h N, ~a i ≤ K h N, ~a i ≤ K independent on the sign of h N, ~a i .Thus, K h N, ~a i ≤ H . From (14) and (17) and since | σ | = 4 H − K , we have Z S | σ | h N, ~a i d S = 4 H Z S h N, ~a i d S − Z S K h N, ~a i d S ≥ H Z S h N, ~a i d S = 2 πr H . The proof then follows the same steps as in the proof of Theorem 2 and we concludethat S is umbilic, and hence a planar disk or a spherical cap. q.e.d We end with a comment. It would be interesting to have a proof of Theorem 1, thatis, the boundary version of the Alexandrov theorem, without invoking the MaximumPrinciple for elliptic equations, as was done by Reilly in [12] for the closed case. Wedo not know a way of applying our arguments to this question.10 eferenceseferences