Stationary liquid drops in Lorentz-Minkowski space 5 0 Rafael L´opez 0 Departmento de Geometr´ıa y Topolog´ıa 2 Universidad de Granada n 18071 Granada (Spain) a J e-mail:[email protected] 3 1 2 Abstract v 8 This paper analyzes the configurations of shapes that shows a space- 3 like liquid drop in Minkowski space deposited over a spacelike plane Π. 0 WeassumethepresenceofauniformgravityfielddirectedtowardΠand 1 that the volume of the drop is prescribed. Our interest are the liquid 0 5 dropsthatarecriticalpoints ofthe energyofthe correspondingmechan- 0 ical system and we will say then that the liquid drop is stationary. In h/ suchcase,theliquid-airinterfaceisdeterminedbytheconditionthatthe p meancurvatureisalinearfunctionofdistance fromΠandthatthe drop - makes a constant hyperbolic angle of contact with the plate Π. As first h result, we shall prove that the liquid drop must be rotationalsymmetric t a with respect to an axis orthogonal to Π. Then we prove the existence m and uniqueness of symmetric solutions for a given angle of contact with : Π. Finally, we shall study the shapes that a liquid drop can adopt in v i terms of its size. So, we shall derive estimates of its height, volume and X area of the wetted surface on Π. r a 1 Introduction and statement of the main results In Lorentz-Minkowski three-dimensional space L3 we are interested for the following physical setting. Consider a liquid drop X of a prescribed volume that is adjacent to a solid surface Σ, called the support surface. We assume that no chemical reaction occurs between the two materials and these ones are homogeneous. We admit the existence of a uniform gravity vector field pointing toward Σ. The energy of the physical system involves the area of the region of contact of X with Σ, (the liquid-solid interface) and the surface area of the drop (the liquid-air interface). For this reason, we imposes in X and 1 Σ a spacelike causal condition that allows to consider areas of surfaces. This leads to that the gravity is a timelike vector field. We seek the configurations that adopt the liquid drop in a state of equilibrium, that is, when the energy of the physical system is critical under perturbations of the system that do not change the amount of liquid of X. The parameters that are determined physically can be: i) the support surface and the angle of contact; ii) the area of the region that wets the drop and the angle of contact, or iii) the volume of the drop and the angle of contact with the support surface. Fromthemathematical viewpoint,wearestudyingthepossibleshapesofa spacelike surface in Minkowski space L3 whosemean curvature is a function of its position in space, and which meets a given spacelike surface in a prescribed hyperbolic angle. The interior of the liquid drop is a boundeddomain X of L3 whose boundary ∂X decomposes into ∂X = Ω, where is the liquid-air S ∪ S interface and Ω = X Σ is the region in Σ occupied by the part of the drop ∩ that wets on Σ. According to the principle of virtual work, and when the equilibrium of the system is achieved, the interface is characterized by the S equation ∆x = 2HN for the position vector x on the free surface ; here ∆ S denotes the Laplace-Beltrami operator on , N is a unit timelike vector field S normal to and H is the mean curvature of . The hyperbolic angle β with S S which andΣintersect along∂ = Σisdeterminedasaphysicalconstant S S S∩ depending only on the materials. Physically, the boundary ∂ corresponds S with the liquid-air-solid interface of the system. When X achieves a state of equilibrium, we shall say then that X is a stationary liquid drop. In a vertical gravity vector field, the Euler-Lagrange equation for the drop interface requires that H be a linear function of the height, that is, H(x) =κx (x)+λ, x , 3 ∈ S wherex indicatesthethirdspacecoordinate. Hereκisthecapillarity constant 3 and, as the angle β, depends on the materials involved. The constant λ is a Lagrange multiplier arising from the volume constraint. On the other hand, when we talk of contact angle, it is implicitly assumed that the boundary regularity of is enough to ensure that the idea of a normal to at every S S boundary point makes sense. For this, we will require to be a sufficiently S smooth surface up to the boundary ∂ . S When κ = 0, no gravity appears in the physical system. In this situation, the liquid drop is modeled by a surface with constant mean curvature H = S λ/2. When the support surface is a plane, then the surface is a planar disc (H = 0) or a hyperbolic cap (H = 0) (see [2] and Theorem 3.1 for another 6 proof). In this sense, our interest is focused for the case that κ= 0. 6 2 Although in this article we shall consider the case that the support sur- face is a spacelike plane, other interesting cases can appear for more general geometric configurations, as for example, a hyperbolic plane, or liquid bridges interconnecting a set of spacelike planes and hyperbolic planes (see the end of Section 3). This paper is divided in seven sections and organized as follows. We begin in Section 2 with a preparatory introduction where we give precise definitions and pose the formulation of the problem variational for our physical system. Next, we consider stationary liquid drops resting on a spacelike plane Π in a vertical gravity field directed toward Π. We generalize what it occurs for sur- faces with constant mean curvature and we shall prove in Section 3, Theorem 3.1: Undertheeffectofthegravity, anystationary liquiddropinMinkowski space L3 resting in a spacelike plane Π is rotationally symmetric with respect to a straight-line orthogonal to Π. Assuming then rotational symmetry, the Euler equation becomes an ordi- nary differential equation for the profile curve that defines the liquid drop. In successive sections, we discuss in some detail the solutions of this differential equation starting in Section 4 for the problem of existence and uniqueness. As consequence, we prove (Theorems 5.2 and 7.2): Let κ be a constant of capillarity. Given a spacelike plane Π and a real number β, there exists a stationary liquid drop in Minkowski space L3 supported onΠandwhere β istheangleofcontactbetween the drop and Π. The drop is unique up isometries of the ambient space L3. As in Euclidean space, there exists a qualitative difference of shapes that adoptadropaccordingtothesignofκ,thatis,sessileliquiddropswhenκ > 0 and pendent liquid drops if κ < 0. In Sections 5 and 6 we analyze the size of the shapes that a sessile liquid drop can adopt by deriving estimates of the height, volume and properties of monotonicity. Finally, Section 7 is devoted to thestudyofpendentdrops. We referto thereaderthese sections for precise statements. These results provide estimates of the drop in terms of prescribed values. For example (Corollary 6.1) Let Π be a spacelike plane in L3. Consider β,R > 0. Then for any stationary sessile liquid drop X resting on Π such that β is 3 the angle of contact and R is the radius of the disc of Π that wets X, the height q of X satisfies: coshβ 1 q < R − , sinhβ We point up also an existence result in terms of the volume (Theorems 6.7 and 7.3) For each constant of capillarity κ, β R and a positive real number ∈ , there exists a unique sessile liquid drop and a unique pendent V drop resting on a spacelike plane Π enclosing a liquid of volume V and that makes a constant angle β along the liquid-air-solid inter- face. We want to point out although much of our techniques are similar than in Euclidean ambient, differences appear in the Lorentzian setting. The main fact is that the spacelike condition is a strong geometric restriction for the possible configurations. It is worthwhile to bring out some of them: 1. Liquid drops in L3 can extend to be graphs in the whole plane. This means that the associated Euler equation can solved be entire solutions. Our drops correspond with physical situations of wetting, but dewetting is prohibited. 2. Sessile liquid drops in L3 have not the phenomenon of meniscus. More- over and by fixing the constant of capillarity, the ambient space L3 can be foliated by sessile drops. 3. Pendent liquid drops in L3 do not present vertical points. In particular, for any given volume, there exists a pendent drop enclosing that volume and that it is physically realizable. There exists an extensive literature relating to liquid drops in Euclidean space. This is due to its interest both in physic and chemistry for any in- terfacial phenomena in the theory of colloids and new materials. From the mathematical viewpoint, we refer to the book of R. Finn [10], which contains abundant bibliography. Finally, we remark that much of the results obtained here can be straightforward generalize for spacelike hypersurfaces in Ln+1. 4 2 Preliminaries and formulation of the variational problem In this section we present the setting for the presence of our stationary liquid drops,aswellas, weshallprecisethemathematical formulation ofthephysical situation. Let L3 denote the 3-dimensional Lorentz-Minkowski space, that is, thereal vector space R3 endowed withtheLorentzian metric , = dx2+dx2 h i 1 2− dx2, where x =(x ,x ,x ) are the canonical coordinates in L3. An immersion 3 1 2 3 x :M L3 of a smooth surface M is called spacelike if the induced metric on → M is positive definite. Observe that~a= (0,0,1) is a unit timelike vector field globally defined on L3, which determines a time-orientation on the space L3. This allows us to choose a unique unit normal vector field N on M which is in the same time-orientation as ~a, and hence that M is oriented by N. We will refer to N as the future-directed Gauss map of M. In this article all spacelike surfaces will be oriented according to this choice of N. The spacelike condition imposes topological restrictions to the immersion x. For example, there are not closed spacelike surfaces and then, any compact spacelike surface has non-empty boundary. If Γ is a closed curve in L3 and x : M L3 is a spacelike immersion of a compact surface, we say that the → boundary of M is Γ if the restriction x : ∂M Γ is a diffeomorphism. For → spacelike surfaces, the projection π : L3 Π = x = 0 , π(x ,x ,x ) = 3 1 2 3 → { } (x ,x ,0) is a local diffeomorphism between int(M) and π(int(M)). Thus, 1 2 π is an open map and π(int(M)) is a domain in Π. The compactness of M implies that π : M Ω is a covering map. Thus, we have → Proposition 2.1 Let x : M L3 be a compact spacelike surface whose → boundary Γ is a graph over an open region Ω x = 0 . Then x(M) is 3 ⊂ { } a graph over Ω. As conclusion, the boundness of our liquid drops means that the liquid-air interface is a graph on the support surface. We suppose that the boundary S of a spacelike compact surface is included in a plane Π. This plane must be of spacelike-type. We point out also that although the boundary ∂M is possibly non-connected,thecausalcharacteronM impliestheexistenceofacomponent of x(∂M), namedΓ , suchthatπ(int(M)) iscontained intheboundeddomain 0 determined by Γ in Π. Therefore, x(M) defines an ”interior” domain, that 0 is, there exists a bounded region Ω Π such that x(M) Ω determines in R3 ⊂ ∪ a bounded domain B, called the ”interior” of M. 5 For spacelike immersions, the notions of the first and second fundamental form are defined in the same way as in Euclidean space. In classical notation, the first and the second fundamental form of x are I = g dx dx , II = h dx dx , ij i j ij i j ij ij X X where g = ∂ x,∂ x is the induced metric on M by x and h = ∂ N,∂ x . ij i j ij i j h i h i Then the mean curvature H of x is given then by h g 2h g +h g 22 11 12 12 11 22 2H = − . det(g ) ij Assume that M is the graph of a smooth function u = u(x ,x ) definedover a 1 2 domain Ω. The spacelike condition implies u < 1, where is the gradient |∇ | ∇ operator in R2 and the Gauss map is ( u,1) N = ∇ . 1 u2 −|∇ | Accordingthisorientation,themeanpcurvatureH atthepoint(x,u(x)),x Ω, ∈ satisfies the equation (1 u2)∆u u u u = 2H(1 u2)3/2. i j ij −|∇ | − −|∇ | X This equation can alternatively be written in divergence form u div(Tu) = 2H, Tu = ∇ . (1) 1 u2 −|∇ | p We presentnow thenotionof stationary surfaceinL3. Thesupport surface Σ is defined as an embedded connected spacelike surface in L3 that divides the space L3 into two connected components. Let us orient Σ by the future- directed unittimelike vector field N andconsider L3 thecomponentof L3 Σ Σ + \ towardsN ispointing. LetM beaconnectedcompactsurfacewithboundary Σ ∂M and x : M L3 a spacelike immersion, smooth even at ∂Σ such that → x(int(M)) L3 and x(∂M) Σ. A variation of x is a differentiable map ⊂ + ⊂ X : ( ǫ,ǫ) M L3 such that X : M L3, t ( ǫ,ǫ), defined by t − × → → ∈ − X (p) = X(t,p), p M, is an immersion and X = x. The variation is called t 0 ∈ admissible if X (int(M)) L3 and X (∂M) Σ for all t. The area function t ⊂ + t ⊂ A :( ǫ,ǫ) R is defined by − → A(t) = dA t ZM 6 where dA is the area element of M in the metric induced by X ; the area t t function S :( ǫ,ǫ) R of the wetted surface on Σ is defined by − → S(t)= dΣ, ZΩt that is, the area of Ω Σ, the region in Σ bounded by X (∂M). Finally, the t t ⊂ volume function V : ( ǫ,ǫ) R is defined by − → V(t) = X dV, ∗ Z[0,t]×M where dV is the canonical volume element of L3. The number V(t) represents the volume enclosed between the surface X and X . The variation X is said t to bevolume-preserving if V(t) = V(0) for all t and thevariational vector field of X is defined on M by ∂X ξ(p)= (p) . ∂t (cid:12)t=0 (cid:12) Moreover, we assume the existence of a po(cid:12)tential energy Y = Y(p), p L3. (cid:12) ∈ The resultant variation energy is Y(t) = Y dA . t ZM The energy function E :( ǫ,ǫ) R is defined by − → E(t) = A(t) coshβ S(t)+Y(t), − where β R is an arbitrary real constant. We say that the immersion x is ∈ stationary if E (0) = 0 for any volume preserving admissible variation of x. ′ For an arbitrary variation, it can be shown that A(0) = 2 H N,ξ dM ν,ξ ds ′ − h i − h i ZM Z∂M S (0) = ν ,ξ ds ′ Σ − h i ZM Y (0) = Y N,ξ dM, ′ h i ZM where ν and ν are the inward-pointing unitary conormal to M and Ω along Σ ∂M respectively. Moreover, the first variation formula of the volume is given by (cf. [3, 4]) V (0) = N,ξ dA. ′ − h i ZM 7 Hence we obtain the first variation formula for the energy of the physical system: E (0) = ( 2H +Y +λ) N,ξ dM + ξ,ν (coshβ+ N,N ) ds ′ Σ Σ − h i h i h i ZM Z∂M Thus, we have Proposition 2.2 Let Σ be a support surface in L3 and let M be a compact surface. Let us consider x : M L3 a smooth spacelike immersion such that → x(int(M)) L3 and x(∂M) Σ. Then x is stationary if and only if ⊂ + ⊂ 1. The mean curvature H of x satisfies the relation 2H(p) = Y(p)+λ, p M, ∈ where Y is a potential energy and λ is a Lagrange parameter determined by an eventual volume constraint; 2. The surface = x(M) meets the support surface Σ in a constant hyper- S bolic angle β, that is, coshβ = N,N along ∂M. Σ −h i In this article, our interest will center on the case for which: 1. The support surface is a spacelike hyperplane Π. After an isometry, we will assume that Π is parallel to the plane x = 0 and 3 { } 2. The vector field Y(p) is a vertically directed gravitational potential to- wards Π, that is, Y(p) =κ x (p)+λ, for constants κ and λ: 3 We know from Proposition 2.1 that a compact drop resting in spacelike plane is the graph of a function u. In such case, 1 N,N = = coshβ along ∂M. Π −h i 1 u2 −|∇ | p Moreover, the constancy of the hyperbolic angle along ∂M means that the Euclidean angle is also constant along this curve since, u is constant along |∇ | ∂M and 1 NE,~a = along ∂M, h iE 1+tanh2β q 8 where NE and , denote, respectively, the Euclidean unit normal of M and E h i the Euclidean metric of R3. Since weshall studystationary liquid drops,throughoutthis work we shall omit the word stationary and it is implied that a liquid drop is a stationary liquid drop. 3 Symmetry under gravitational fields In this section we prove that the equilibrium shape of a drop of liquid in L3 resting over a spacelike plane in a uniform gravitational field is rotational symmetric with respect to a straight-line orthogonal to the support surface. Exactly we show Theorem 3.1 Let M be a spacelike embedded compact connected surface in L3 whose boundary ∂M is contained in a plane Π. Assume that M lies in one side of Π and the two following assumptions hold: 1. The mean curvature H of M depends only on the distance to Π. 2. The hyperbolic angle of contact between M and Π is constant along ∂M. Then M is rotational symmetric with respect to a straight-line orthogonal to Π. Moreover, M is a topological disc. This result is analogous to it happens in Euclidean ambient and which was proved by Wente [22]. It turns out that the method of proof used there, called the Alexandrov reflection technique, may be adapted to the present situation. Such technique was firstly used to prove that a closed embedded constant mean curvature surface in R3 must be a round sphere [1]. See also the remarkable reference[20]in the context of thetheory ofpartial differential equations. The proof idea is to use the very surface M as comparison surface with itself and to apply the Hopf maximum principle for elliptic equations. We consider ui, i = 1,2, two functions satisfying div(Tui) = 2H (x), ui < 1 i |∇ | in a domain Ω, with H (x) H (x). The operator div(Tu) may be written 1 2 ≤ in the form (1 u2)u +2u u u +(1 u2)u div(Tu) = − 2 11 1 2 12 − 1 22, W = 1 u2 W3 −|∇ | q 9 where the subscript i indicates the differentiation with respect to the variable x . We write div(Tu) = a (x,u, u)u with p = (p ,p ), p = u , and i ij ij 1 2 i i ∇ where P (1 p 2)ξ 2+ ξ,p 2 a (x,u,p)ξ ξ = −| | | | h i . ij i j W3 X Then it holds 0< λ(x,u,p)ξ 2 a (x,u,p)ξ ξ Λ(x,u,p)ξ 2, ij i j | | ≤ ≤ | | X 1 1 λ(x,u,p) = Λ(x,u,p) = . W W3 Then the operator is elliptic for p < 1 and uniformly elliptic for compact | | domains. Let φ(x,p,r) = div(Tu) = 2H(x) (2) where r = (r ), r = u . Then φ is a smooth function defined in Ω D R4 ij ij ij × × given explicitly by 1 p p i j φ(x,p,r) = δ + r , 1 p 2 ij 1 p 2 ij −| | X(cid:18) −| | (cid:19) p where D is the unit open disc of R2. For each u= ui, i= 1,2, we will use the notation pi,ri and H for each i . Since H H , a standard argument using i 1 2 ≤ the chain rule shows then 0 φ(x,p2,r2) φ(x,p1,r1) ≤ − 1 ∂φ 1 ∂φ = (θ(t)) dt w + (θ(t)) dt w := Lw, ij j ∂r ∂p Z0 ij Z0 j X X where w = u1 u2, w = ∂w/∂x , w = ∂2w/∂x ∂x and θ = θ(t)= (x,tp2+ i i ij i j − (1 t)p1,tr2 +(1 t)r1). The right hand side of the above equation defines − − an elliptic operator L because 1 1 1 1 ξ 2 dt ξ 2 Lw max , ξ 2, | | ≤ W(θ(t)) | | ≤ ≤ W3 W3 | | Z0 (cid:26) 1 2 (cid:27) and W = 1 ui 2. Since the coefficients a are locally bounded, L is i ij −|∇ | locally unifoqrmly elliptic and we are in position to apply the Hopf’s maximum principletothedifferencefunctionw ([12]; seealso[11,Ch. 3]). Consequently, we have proved the following result. 10