Equilibrium of Surfaces in a Vertical Force Field

Mediterr. J. Math. (2022) 19:3 https://doi.org/10.1007/s00009-021-01877-4 c The Author(s) 2021 Equilibrium of Surfaces in a Vertical Force Field Antonio Mart´ınezandA.L.Mart´ınez-Trivi˜no Abstract. In this paper, we study ϕ-minimal surfaces in R3when the function ϕis invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in R2.We describe a full classification of complete flat-embedded ϕ-minimal surfaces if ϕis strictly monotone and characterize rotational ϕ-minimal surfaces by its behavior at infinity when ϕhas a quadratic growth. Mathematics Subject Classification. 53C42, 35J60. Keywords. ϕ-minimal, elliptic equation, weighted volume functional. 1. Introduction The equilibrium of a flexible, inextensible surface Σ in a force field F= (X,Y,Z)ofR3, was given by Poisson [20, pp. 173–187] and when the intrinsic forces of the surface are assumed to be equal, the external force must have a potential Twhich corresponds, up to a constant, with the tension of the surface, that is, dT+Xdx+Ydy+Zdz=0.(1.1) In this case, the equilibrium condition is given in terms of the mean curvature vector Hof Σ as follows: HT+F⊥= 0 (1.2) where ⊥denotes the projection to the normal bundle of Σ. From equations (1.1) and (1.2), Poisson obtains: •The minimal surface equation, by taking F=0andT=const. •The capillary surface equation, by taking T=const and Fnormal to the surface with F depending linearly on the height. The authors were partially supported by MICINN-FEDER, Grant No. MTM201680313-P and Junta de Andaluc´ıa Grant No. FQM325. 0123456789().: V,-vol 3 Page 2 of 28 A. Martínez and A. L. Martínez-Triviño MJOM •The equation of a heavy surface in a gravitational field, by taking T= (0,0,gE(z)), g= gravitational constant and E(z) a density function on the surface. In this paper, we are interested in the last case, that is, when the equation (1.2) gives H=(∇ϕ)⊥=˙ϕe⊥ 3,(1.3) where ϕ(z) = log z z0gE(t))dt,∇is the gradient operator in R3and(˙) denotes derivate respect to the third coordinate. To get a regular problem, we have to restrict the surfaces to the region of R3where ϕis regular. These surfaces are a particular case of the so called f-minimal surfaces (see [3]) for which the function fdepends only on the height. They can be viewed either as critical points of the weighted volume functional Vϕ(Σ) := Σ eϕdAΣ,(1.4) where dAΣis the volume element of Σ, or as minimal surfaces in R3with the conformally changed metric Gϕ:= eϕ·,·.(1.5) From this property of minimality, a tangency principle can be applied and any two different ϕ-minimal surfaces cannot “touch” each other at one interior or boundary point (see [7, Theorem 1 and Theorem 1a]). Any surface satisfying (1.3) will be called [ϕ, e3]-minimal and if Σ is the vertical graph of a function u:Ω⊆R2−→ R, we also refer to uas [ϕ, e3]- minimal. Hence, uis [ϕ, e3]-minimal if and only if it solves the following [ϕ, e3]-minimal equation: (1 + u2 x)uyy +(1+u2 y)uxx −2uyuxuxy =˙ϕ(u)1+u2 x+u2 y.(1.6) This kind of surfaces has been widely studied specially from the viewpoint of calculus of variations. Classical results about the Euler equation and the existence and regularity for the solutions of the Plateau problem for (1.4) can be found in [2,9–11,24]. But contributions from a more geometric viewpoint only has been given for some particular functions ϕ. It is interesting to mention •The case of ϕ(z)=z: it corresponds with translating solitons, that is, surfaces in R3such that t→ Σ+te3 is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point: H=e⊥ 3. Recent advances in the understanding of its local and global geometry can be found in [4,8,12–14,16,17,23,25] •The case of ϕ(z)=αlog z,α=const. It includes the two dimensional examples analogues of the catenaries (when α= 1). We refer to [2,5,6, 15,19] for some progress made in this family. MJOM Equilibrium of Surfaces in a Vertical Force Field Page 3 of 28 3 The aim of this paper is develop a general and systematic approach to study [ϕ, e3]-minimal surfaces from a geometric viewpoint. Nonetheless, the class of [ϕ, e3]-minimal surfaces is indeed very large and much richer in what refers to examples and geometric behaviors. Although new ideas are needed for its study, it will be necessary, to get classification results, to impose some additional conditions to the function ϕ. Here, as a general assumption, we will always consider ϕstrictly monotone, that is, ϕ:]a, b[⊆R→Ris a strictly increasing (or decreasing) function and Σ ⊂R2×]a, b[.(1.7) Invariant surfaces by an uniparametric group of rigid motions in R3are related with the one-dimensional case of (1.6). Since ϕis taking so arbitrary, we only consider [ϕ, e3]-minimal surfaces invariant by two types of uniparametric groups, namely, groups of horizontal translations and the group of vertical rotations. In the first case, besides vertical planes, we may consider that u=u(x), x∈Idepends only on x. Then, from (1.6), the generalized cylinder Σ = {(x, y, u(x)) |x∈I,y ∈R}is a [ϕ, e3]-minimal surface if and only if u satisfies u(x)= ˙ϕ(u)(1 + u(x)2) (1.8) From its physical interpretation, any solution of (1.8) will be called ϕ-catenary. The corresponding generalized cylinder is called [ϕ, e3]-catenary cylinder. If we rotate around the x-axis a [ϕ, e3]-catenary cylinder an angle θ∈]0,π/2[ and dilate by 1 cos θ, the resulting surface is also [ϕ, e3]-minimal and we will say it is a tilted [ϕ, e3]-catenary cylinder. In Theorem 3.7,weprove that any complete flat [ϕ, e3]-minimal surface is either a vertical plane or a [ϕ, e3]-catenary cylinder (maybe tilted). In the second case, we consider [ϕ, e3]-minimal surfaces that are invariant under the one-parameter group of rotations that fix the e3direction. From (1.6), the arc-length parametrized generating curve γ(s)=(x(s),0,z(s)),s∈I⊂R of a such surface satisfies ⎧ ⎨ ⎩ x=cos(θ) z=sin(θ), θ=˙ϕ(z)cos(θ)−sin(θ) x. (1.9) In Theorems 4.5 and 4.11 , we establish the geometric properties of the rotational [ϕ, e3]-minimal surfaces according two types of surfaces: one is globally convex with only one complete embedded end (it is called a [ϕ, e3]-minimal bowl) and the other has two complete embedded convex ends and has a generating curve of winglike type (it is called [ϕ, e3]-minimal catenoid) Very little is known about the geometry of the immersed [ϕ, e3]-minimal surfaces and most of the results have been proved only for translating solitons. One of the first result in that direction was obtained by Clutterbuck, Schn¨ure, Schulzein[4], where they proved that when ˙ϕ≡1, any rotationally symmetric 3 Page 4 of 28 A. Martínez and A. L. Martínez-Triviño MJOM solution u=u(r), r=x2+y2, on the exterior of a compact planar domain has de following asymptotic behavior: u(r)=r2 2−log r+O(r−1). Somewhat later Martin–Savas–Smoczyk proved in [17] that any complete translating soliton with a single end asymptotic to a translating paraboloid is a translating paraboloid. In this paper, we generalize the above results to [ϕ, e3]-minimal with ˙ϕ satisfying the following expansion at infinity: ˙ϕ(u)=αu +β+ ∞  n=1 an un,a n∈R,(1.10) where either α>0 and the first non-vanishing akis positive or α=0, β>0 and the first non-vanishing akis negative. The results we prove can be summarized in the following two theorems Theorem A. If ˙ϕsatisfies (1.10), then any rotationally symmetric solution u of (1.6)has the following asymptotic behavior: •If α>0, ϕ(u)(r)=Ce αr 2+O(r2),C>0,(1.11) •If α=0and up to a constant, we have G(u)(r)=r2 2−1 β2log(r)+O(r−2),(1.12) where Gis the strictly increasing function given by G(u)=u u0 dξ ˙ϕ(ξ). Theorem B. Let Σbe a complete properly embedded [ϕ, e3]-minimal surface in R3with a single end that is smoothly asymptotic to a [ϕ, e3]-minimal bowl, ˙ϕsatisfying (1.10). Then, the surface Σis a [ϕ, e3]-minimal bowl. The paper is organized as follows: in Sect. 2, we show some fundamental equations related to our family of surfaces and as a consequence, we prove the non-existence of closed examples and two results about strictly convexity and mean convexity of [ϕ, e3]-minimal surfaces. Section 3is devoted to the study and classification of embedded complete flat [ϕ, e3]-minimal surfaces. We describe geometrically the so called [ϕ, e3]-catenary cylinders and tilted [ϕ, e3]-catenary cylinders and characterize them together to vertical planes as the unique examples of complete flat [ϕ, e3]-minimal surfaces. In Sect. 4, we study the existence and classification of rotational examples. We construct for ϕin a very general class of functions (strictly increasing and convex) a family of [ϕ, e3]-minimal bowls (which are strictly convex graphs) and [ϕ, e3]-minimal catenoids with a winglike shape (which resemble the usual translating catenoids in R3). MJOM Equilibrium of Surfaces in a Vertical Force Field Page 5 of 28 3 Finally, Sects. 5and 6are devoted to study [ϕ, e3]-minimal surfaces when ϕhas a quadratic growth. We provide the asymptotic behavior of rotationally symmetric examples and characterize [ϕ, e3]-minimal bowls by their behavior at infinity. 2. Some Relevant Equations Here, we will give some local fundamental equations related to [ϕ, e3]-minimal surfaces. Let ψ:M−→ R3be a 2-dimensional [ϕ, e3]-minimal immersion (maybe with a non empty boundary) with Gauss map N, induced metric g and second fundamental form A. We shall denote by ∇,Δand∇2, respectively, the Gradient, Laplacian and Hessian operators of g. The mean curvature vector of ψis defined by H=trace gAand the symmetric bilinear form Agiven by A(X,Y )=−A(X,Y ),N,X, Y ∈TΣ, is called scalar second fundamental form. The mean curvature function Hwill be the trace of Awith respect to g. With this notation, (1.3) is equivalent to H:= −˙ϕN, e3.(2.1) We will assume that ϕsatisfies (1.7) and let us introduce the height and angle functions, respectively, by μ:= ψ, e3,η:= N, e3. Lemma 2.1. The following relations hold: ∇μ=e 3,∇η, ·=−A(∇μ, ·),(1) ˙ϕ2=˙ϕ2|∇μ|2+H2,(2) ˙ϕ∇2μ=HA,(3) ∇2η=(∇A)(∇μ, ·,·)+H ˙ϕA[2],(4) Δμ=˙ϕ(1 −|∇μ|2),(5) ΔN+˙ϕ∇η+¨ϕη∇μ+|A|2N=0,(6) ∇2H=−η∇2˙ϕ−(∇A)(∇ϕ, ·,·)−HA[2] +B(7) ΔA+(∇A)(∇ϕ, ·,·)+η∇2˙ϕ+|A|2A−B=0,(8) where A[2] and Bare the symmetric 2-tensors given by the following expressions: A[2](X,Y )= kA(X,Ek)A(Ek,Y), B(X,Y )=∇ ˙ϕ, X, A(∇μ, Y )+∇ ˙ϕ, Y A(∇μ, X), for any vector fields X,Y ∈TΣand any orthonormal frame {E1,E 2}of TΣ. Proof. (1) Differentiating μand ηrespect to any X∈TΣ, we get ∇μ, X=dμ(X)=e 3,X, ∇η,X=dη(X)=dN(X),e 3=A(X, e 3). 3 Page 6 of 28 A. Martínez and A. L. Martínez-Triviño MJOM (2) From (2.1) and (1), it is clear that 1=|∇μ|2+H2 ˙ϕ2. (3) From definition of the Hessian operator, ∇2μ(X,Y )=XY(μ)−(∇XY)(μ)=A(X,Y ),e 3=−A(X,Y )η. Therefore, (3) follows from (2.1). (4) From Codazzi equation and (2.1): ∇2η(X,Y )= k (∇A)(Ek,X,Y)Ek(μ)− kA(X,Ek)A(Y,Ek)η =(∇A)(∇μ, X, Y )+H ˙ϕA[2](X,Y ). (5) From (2) and (3), Δμ= k∇2μ(Ek,E k)=H2 ˙ϕ=˙ϕ(1 −|∇μ|2). (6) As H=−˙ϕη,wehave ∇H=−¨ϕη∇u−˙ϕ∇η, and (6) follows from the well-known fact that ΔN=∇H−|A|2N. (7) From (2.1) and (4), we obtain ∇2H(X,Y )=XY(H)−(DXY)H =−η∇2˙ϕ(X,Y )−˙ϕ∇2η(X,Y )−∇˙ϕ, Y X,∇η−∇˙ϕ, XY,∇η =−η∇2˙ϕ(X,Y )−(∇A)(∇ϕ, X, Y )−HA[2](X,Y )+B(X,Y ). which give the proof of (7). (8) Using the well-known Simon’s identity: ΔA=∇2H−|A|2A+HA[2] and (7) we obtain (8).  From this Lemma, we have Corollary 2.2. If ϕ:]a, b[→R, is a strictly increasing (or decreasing)function, then the height function μof ψcannot attain a local maximum (or local minimum)at any interior point. Corollary 2.3. There is no any closed 2-dimensional [ϕ, e3]-minimal immersion ψ:M−→ R2×]a, b[. About the sign of the curvatures of ψ,wehave Theorem 2.4. Let ϕ:]a, b[→Rbe a strictly increasing function satisfying ¨ϕ+λ˙ϕ2≥0,for some constant λ>0,(2.19) and let ψ:Σ−→ R2×]a, b[be a 2-dimensional [ϕ, e3]-minimal immersion with H≤0.IfHvanishes anywhere, then Hvanishes everywhere and ψ(Σ) lies in a vertical plane. MJOM Equilibrium of Surfaces in a Vertical Force Field Page 7 of 28 3 Proof. Using (2.1)andtheEqs.(1), (2), (5) and (6) in Lemma 2.1,wehave Δ(e−λϕ)+λe−λϕ(¨ϕ|∇μ|2+H2−λ˙ϕ2|∇μ|2)=0, Δη+˙ϕ∇η,∇μ+(|A|2+¨ϕ|∇μ|2)η=0. Thus, we obtain Δ(e−λϕη)+(2λ+1)∇(e−λϕη),∇ϕ= =−ηe−λϕ((λ+ 1)( ¨ϕ+λ˙ϕ2)|∇μ|2+λH2+|A|2). But, by hypothesis, ηis a nonnegative function, and so, from the strong maximum principle, if it vanishes anywhere then it vanishes everywhere, which concludes the proof.  Theorem 2.5. Let ϕ:]a, b[→Rbe a strictly increasing function satisfying ... ϕ≤0,andletψ:Σ−→ R2×]a, b[be a 2-dimensional locally convex [ϕ, e3]- minimal immersion. If the Gauss curvature Kvanishes anywhere, then K vanishes everywhere. Proof. By hypothesis, the Gauss map Ncan be chosen such that Ais a positive semi-definite bilinear form and from (8), we have ΔA+(∇A)(∇ϕ, . , . )+G(A)=0 where G(A)=η∇2˙ϕ+|A|2A−B. But, from Lemma 2.1,if... ϕ≤0, we obtain G(A)(v,v)=η... ϕ∇μ, v2≤0 for each null vector vof A. Therefore, can apply the maximum principle of Hamilton (see [21, Section 2]) and if there is an interior point of Σ where A has a null-eigenvalue then Amust have a null-eigenvalue everywhere, which concludes the proof of the theorem.  3. Complete Flat [ϕ, e3]-Minimal Surfaces 3.1. Vertical Graphs Invariant by Horizontal Translations Consider the [ϕ, e3]-minimal vertical graph given by a function uwhich only depend on one variable, u=u(x), from (1.6)umust be a solution of the following ODE: u(x)= ˙ϕ(u)(1 + u(x)2) (3.1) To look for complete examples, we will consider that ϕ:]a, ∞[−→ R is either a strictly increasing (or decreasing) function. Then, by taking z= ϕ(u)andu= tan(v), we obtain that (3.1) is equivalent to v=h(z), z=h(z) tan(v),(3.2) where h(z)= ˙ϕ(ϕ−1(z)). 3 Page 8 of 28 A. Martínez and A. L. Martínez-Triviño MJOM Figure 1. Phase portrait of (3.2) It is clear that ezcos(v) is constant along the solutions of (3.2)andfrom Fig. 1, for each solution uof (3.1) there exists a unique x0∈Rsuch that v(x0) = 0 (it is not a restriction to assume that x0= 0). By taking the initial conditions u(0) = u0,u (0) = 0,(3.3) we have that for each x≥0, u(x) is given by u(x):=(X◦ϕ)−1(x),with X(z)=z z0 dτ |h(τ)|√e2(τ−z0)−1,(3.4) where z0=ϕ(u0). Thus, from (3.1) and (3.3), we obtain, Proposition 3.1. The solution uof (3.1)–(3.3)is even and it is defined in the interval ]−Λu0,Λu0[,where Λu0= lim u→∞ ϕ(u) ϕ(u0) dτ |h(τ)|√e2(τ−z0)−1.(3.5) Theorem 3.2. If ϕ:]a, ∞[−→ Ris a strictly increasing function, then, •Λu0<∞if and only if ∞ u0e−ϕ(λ)dλ<∞. Therefore, if Λλ0<∞for some λ0∈]a, ∞[, then Λλ<∞for all λ∈]a, ∞[. •If Λλ<∞and ˙ϕis increasing (respectively, decreasing), then Λλis decreasing (respectively, increasing) in λ. MJOM Equilibrium of Surfaces in a Vertical Force Field Page 9 of 28 3 Proof. As lim τ→∞ √e2(τ−z0)−1 eτ−z0=1=0, the first item follows from (3.5). On the other hand, by assuming that ˙ϕis increasing and Λλ<∞for all λ∈]a, ∞[, we have from (3.5), that, if λ1≤λ2, Λλ1≥Λλ2+ lim z→∞ z−ϕ(λ1) z−ϕ(λ2) dτ h(τ+ϕ(λ1))√e2τ−1=Λ λ2. A similar discussion can be done when ˙ϕis decreasing.  From (3.1), (3.2), (3.3), (3.4), (3.5) and Theorem 3.2,wecanprovethe following properties of the solutions, Theorem 3.3. Let ϕ:]a, ∞[−→ ]b, c[,a, b ∈R∪ {−∞},c∈R∪{∞}be a strictly increasing diffeomorphism, then the solution uof (3.1)–(3.3)is defined in ]−Λu0,Λu0[,Λu0∈{R+,∞}, it is convex, symmetric about the y-axis and has a minimum at x=0. Moreover, •if c<∞, then Λu0=∞,and lim x→±∞ u(x)=∞,lim x→±∞ u(x)=±e2(c−z0)−1. •if c=∞, lim x→±Λu0 u(x)=∞,lim x→±Λu0 u(x)=±∞. In particular, if Λu0<∞, the graph of uis asymptotic to two vertical lines. Theorem 3.4. Let ϕ:]a, ∞[−→ ]b, c[,a, b ∈{R,−∞},c∈{R,∞} be a strictly decreasing diffeomorphism, then the solution uof (3.1)–(3.3)is defined in ]−Λu0,Λu0[,Λu0∈{R+,∞}, it is concave, symmetric about the y-axis and has a maximum at x=0. Moreover, •if c<∞, then Λu0<∞,and lim x→±Λu0 u(x)=a, lim x→±Λu0 u(x)=±e2(c−z0)−1. •if c=∞, then Λu0<∞⇐⇒u0 a e−ϕ(λ)dλ<∞, and lim x→±Λu0 u(x)=a, lim x→±Λu0 u(x)=±∞. Remark 3.5.In the hypothesis of Theorem 3.4, the graph of uis complete when a=−∞. But in this case, by changing ϕby −ϕ, we can also apply Theorem 3.3. Definition 3.6. For each solution uof (3.1)–(3.3), we refer C:= Graph(u)×R as a [ϕ, e3]-catenary cylinder surface (Fig. 2). 3 Page 16 of 28 A. Martínez and A. L. Martínez-Triviño MJOM ]x0,ω +[, such that lim x→ω+ u(x)=∞. For studying the left branch of γ, we are going to consider, γ−(s)= γ(−s)fors∈[0,s −[. Then, by taking x(s)=x(−s), z(s)=z(−s)and θ(s)=θ(−s)+πfor s∈[0,s −[, we have that {x, z,θ}is a solution of (4.8) on [0,s −[ satisfying x(0) = x0>0, z(0) = z0∈]a, ∞[, θ(0) = π. (4.14) Lemma 4.7. There exists s0∈]0,s −[such that 2θ(s0)=π. Proof. Assume on the contrary, θ(s)∈]π 2,π[ for all s∈]0,s −[, and from (4.8)– (4.14), we have that x<0, θ<0andz>0on]0,s −[. Hence, there exist x−= lim s→s− x(s), z−= lim s→s− z(s), θ−= lim s→s− θ(s), and as ] −s−,s +[ is the maximal interval of existence of γ,wehavethat either x−=0orz−=∞. Therefore, γ−is the graph of a convex function u=u(x)on]x−,x 0[ such that either x−= 0 or limx→x−u(x)=+∞. In the first case, if limx→x−u(x)=+∞, from the convexity of uwe get that θ−=π 2and there exists a sequence {sn}→s−satisfying θ(sn)→0, but then, from (4.8), 0 = lim n→∞ θ(sn) = lim n→∞ cos(θ(sn)) ˙ϕ(z(sn)) −sin(θ(sn)) x(sn) ≤lim n→∞ cos(θ(sn)) ˙ϕ(z(sn)) ≤0. Thus, 0 = lim n→∞ cos(θ(sn)) ˙ϕ(z(sn)) = lim n→∞ sin(θ(sn)) x(sn)=1 x−=0, which is a contradiction. If x−= 0 then, from [22, Theorem 2], limx→0u(x)=+∞and arguing as above we also obtain a contradiction.  Lemma 4.8. If s∈]s0,s −[, then 0<2θ(s)<π. Proof. It is clear because θ<0onθ−1(π 2)andθ>0onθ−1(0).  Lemma 4.9. θhas a minimum at a point s1∈]s0,s −[and θ>0on ]s1,s −[ Proof. Assume that θ<0on]s0,s −[. Then, from Lemma 4.8,xx−, zz−and θθ−∈[0,π 2[ when s→s−. In particular, there is a sequence {sn}→s−satisfying limn→∞ θ(sn)=0. Under this assumption, we assert that θ−=0andx−<+∞, otherwise 0 = lim n→∞ θ(sn) = lim n→∞ cos(θ(sn)) ˙ϕ(z(sn)) −sin(θ(sn)) x(sn) = lim n→∞ cos(θ−) lim n→∞ ˙ϕ(z(sn)) ≥cos(θ−)˙ϕ(z0)>0, MJOM Equilibrium of Surfaces in a Vertical Force Field Page 17 of 28 3 which is a contradiction. Thus, γ−is the graph of a concave function u=u(x) on a bounded interval ]x(s0), x−[ satisfying limx→x−u(x)=+∞but this is also a contradiction because θis strictly decreasing on ]s0,s −[. Hence, there exists s1∈]s0,s −[ such that θ(s1) = 0. Moreover, from (4.8), θ(s1)=sin(θ(s1)) cos(θ(s1))( ¨ϕ(z(s1)) + 1 x2(s1))>0, and s1is a local minimum of θ. Now, arguing as in Theorem 4.5,wecan prove that, on the interval ]x(s1), x−[, γ−is the graph of a convex function satisfying lim x→x− u(x)=+∞.  Lemma 4.10. The profile curve γis embedded. Proof. Let s0∈]−s−,0[ the point given by the Lemma 4.7 and consider the following branches of γdetermined by γ−s−,s0[and γ]s0,s+[, respectively, parametrized by u+(x)=(x, u+(x)) for any x∈]x0,x(s−)[ u−(s)=(x, u−(x)) for any x∈]x0,x(s+)[, where uis solution of the Eq. (4.12). Now, define the following smooth function d(x)=u +(x)−u −(x). It is clear that d(x)>0forx∈]x0,x 0+δ[for some δ>0. Suppose that there exists a first r≥x0+δsuch that d(r)=0 and d(r)≤0. Consequently, u+(x)>u −(x) for any x∈]x0,r[ and from the equation 4.12, we get to contradiction since, d(r)=(1+u(r)2)(˙ϕ(u+(r)) −˙ϕ(u−(r))) >0. Thus, d>0 everywhere and integrating u+(x)>u −(x) for any x≥x0. Theorem 4.11. For every x0>0, there exists a complete embedded rotational [ϕ, e3]-minimal, see Fig. 5(right)with the annulus topology whose distance to axis of revolution is x0and whose generating curve γis of winglike type see Fig. 5(left). These examples will be called [ϕ, e3]-minimal catenoids. Proof. It follows from Lemmas 4.7,4.8,4.9 and 4.10. Proposition 4.12. Under the above conditions, the following statements hold: 1. If ˙ϕhas at most a linear growth, then ω+=+∞and x−=+∞. 2. If ˙ϕgrowths as uαfor some α>1, then ω+, x−∈R. Proof. If ˙ϕhas at most a linear growth, then there must be a constant c>0 such that ˙ϕ(u)/u ≤coutside a compact set. Thus, from the equation (4.12), when xis large enough the following inequalities hold: x≥u ˙ϕ(u)(x)≥1 c u u(x).(4.15) 3 Page 18 of 28 A. Martínez and A. L. Martínez-Triviño MJOM 0 5 10 15 20 0 5 10 15 20 25 Figure 5. [ϕ, e3]-minimal catenoid with ˙ϕ(u)=e−1/u Integrating both members of the inequality (4.15), we get that x2 2−x2 0 2≥1 clog u(x) u(x0)for some x0>0.(4.16) Hence, ω+=+∞and x−=+∞. Let us go to consider now that lim u→+∞ ˙ϕ(u) uα=M= 0 for some α>1, and suppose that ω+=+∞. Then, from the Theorems 4.5 and 4.11,thereal function fgiven by f(r):= u(r) Mu α(r) has, for rlarge enough, a bounded and strictly monotone primitive F(u)(r). Hence, there exists a sequence {rn}+∞such that lim n→∞ f(rn)=0.(4.17) Claim 4.13. The function fsatisfies that limr→∞ f(r) r=0. ProofofClaim4.13.Assuming on the contrary, there exists δ>0 and a sequence {sn}+∞such that f(sn)>f(sn) sn >δ, which together (4.17), says that f−1(δ) is unbounded real subset containing a divergent sequence to +∞. MJOM Equilibrium of Surfaces in a Vertical Force Field Page 19 of 28 3 But, from the equation (4.1), the function fsatisfies the following differential equation: f=˙ϕ(u) Mu α−f(r) r+M2f2u2α˙ϕ(u) Mu α−f r−α Mu−α+1(4.18) and we obtain that there exists ˆr∈f−1(δ) such that f(r)>1 for any r∈f−1(δ), r≥ˆr, which is impossible because f−1(δ) is unbounded.  From (4.18), Claim 4.13 and using that udiverges to +∞we get that, for rsufficiently large, the following inequality holds: 2f 1+f2>1.(4.19) By integration of this expression, we conclude that ω+<+∞. Remark 4.14. Notice that ω+=+∞does not imply that ˙ϕhas at most a linear growth. For example, by taking ˙ϕ(u)=ulog(u) with u≥1 and by the integration of both members in (4.15), we get that x2 2−x2 0 2≥log log u(x) u(x0) for some x0>0. Thus, ω+=+∞but the function log(u) is not bounded. 5. Asymptotic Behavior of Rotational Examples Clutterbuck, Schn¨urer and Schulze studied in [4] the asymptotic behavior of solitons rotationally symmetric. They proved that the problem u=(1+u2)1−u r,r>R, u(R)=u0∈R,u (R)=u1∈R.(5.1) has a unique C∞-solution uon [R, ∞[. Moreover, as r→∞,uhas the following asymptotic expansion: u(r)=r2 2−log(r)+O(r−2). Due to the arbitrariness of the problem (4.1), it is impossible to find a general asymptotic behavior of their solutions because if you consider any strictly convex smooth function u=u(r), r>R, one can find a function ϕ such that uis a solution of (4.1). Proposition 4.12 motivates to consider ϕ:]a, +∞[−→ Ra smooth function satisfying (4.9) and with a quadratic growth, that is, with the following asymptotic behavior: lim u→∞ ¨ϕ(u)=α≥0 and lim u→∞(˙ϕ(u)−αu)=β∈R.(5.2) In this case, we are going to generalize the result in [4] to the following problem: u=(1+u2)˙ϕ(u)−u r,r>r 0≥0, u(r0)=u0>a, u (r0)=u1≥0,(5.3) 3 Page 20 of 28 A. Martínez and A. L. Martínez-Triviño MJOM with ϕ:]a, ∞[−→ Rsatisfying (4.9) and (5.2). Remark 5.1.Observe that if α>0, then uis solution of (4.1) if and only if v=u+β− β αis solution of v=(1+v2)˙ ψ(v)−v r where ψ(v)=ϕv−β− β αsatisfies lim v→∞ ¨ ψ(v)=α≥0 and lim v→∞(˙ ψ(v)−αv)= β. It is also clear that v ˙ ψ(v)=u ˙ϕ(u). Theorem 5.2. (Case α>0) Assume that ˙ϕ(u0)r0≥u1and α>0.Then, the problem (5.3)has an unique strictly convex C∞-solution uon [r0,∞[. Moreover, as r→∞, we have the following asymptotic expansion: ˙ϕ(u)(r)=e1 2αr2+o(r2)(5.4) u ˙ϕ(u)(r)=r−αr ˙ϕ(u)−2(r)+or˙ϕ(u)−2(r),(5.5) Proof. First of all, arguing as in Theorems 4.5,4.11 and Proposition 4.12 , (5.3) has a unique C∞-solution uon [r0,∞[ which is strictly convex function satisfying that limr→∞ u(r)=∞. Hence, from (4.1), r˙ϕ(u)>u,r≥r0.(5.6) From Remark 5.1, to study the asymptotic behavior of u ˙ϕ(u),itisnotarestriction to assume that β>0. Take >0 such that β>2ε,from(5.2) there exists rεsuch that if r≥rε, −ε< ˙ϕ(u)(r)−αu(r)−β<ε, −ε< ¨ϕ(u)(r)−α<ε. (5.7) Lemma 5.3. Consider for any R>r 0, the function ζR(r):=gεu(R)+r R t˙ϕ(u)(t)dt,r≥R, gε=β−2ε β+ε. Then, there exists r1∈R, depending only on ε, such that for any R≥r1,ζR satisfies the following inequality: ζ R<(1 + ζ2 R)˙ϕ(ζR)−ζ R r,r≥R. (5.8) Proof. From the inequality (5.6), ζR(r)>u(r)gε. Hence, from (5.7), when r is large enough, we have ˙ϕ(ζR)(r)>αg εu(r)+β−ε. (5.9) Using (5.6), (5.7) and by a straightforward computation, ζ R(r)<g ε˙ϕ(u)(r)(1 + (α+ε)r2),r≥rε,(5.10) MJOM Equilibrium of Surfaces in a Vertical Force Field Page 21 of 28 3 On the other hand, from (5.9) and (5.7), when r≥rε, the following inequality holds: (1 + ζ2 R)˙ϕ(ζR)−ζ R r>ε(1 + ˙ϕ(u)2r2g2 ε).(5.11) Thus, (5.8) follows from (5.9), (5.10), (5.11) bearing in mind that u→+∞ when r→+∞. Lemma 5.4. For any R≥r0there exists rR≥Rsuch that u(rR)−ζ R(rR)> 0. Proof. Assuming on the contrary, if u(r)−ζ R(r)≤0 for any r>R, then the following inequalities holds: u(r) 1+u2(r)≥3ε β+ε˙ϕ(u)(r)>3ε β+ε˙ϕ(u)(r0), Integrating, we can find a finite radius rsuch that u→+∞as r→r, getting acontraction since the solution uis defined for all r>r 0. Let us consider the function d=u−ζ Ron [R, ∞[. From Lemmas 5.3 and 5.4 , we can find Rr0verifying u(R)>0, d(R)>0 and such that the inequality (5.8) holds. Hence, if there exists a first s≥Rsuch that d(s)=0 and d(s)<0, we have 0>d (s)=(1+u(s)2)( ˙ϕ(u(s)) −˙ϕ(ζR(s))). On the other hand, as d(r)>0 for any r∈]R, s[, we have by integration of dthat u(s)>ζ R(s)+u(R)−ζR(R)=ζR(s)+ 3ε β+εu(R)>ζ R(s), and (4.9) gives that d(s)>˙ϕ(u(s)) −˙ϕ(ζR(s)) >0 which is a contradiction. Thus, d(r)>0forrlarge enough and using the inequality (5.6), we get u(r) ˙ϕ(u)(r)=r+V1(r),with lim r→+∞V1(r) r=0.(5.12) Moreover, from the previous formula (5.12)andL’Hˆopital’s rule, we also get that lim r→+∞ log ˙ϕ2(u(r)) αr2=1 and ˙ϕ(u) has the following asymptotic expansion: ˙ϕ(u)(r)=e1 2αr 2+o(r2).(5.13) Lemma 5.5. V1→0as r→+∞. Proof. As V1is sublinear, we have that for rlarge enough, |V1(r)|<crfor all c>0. Moreover, from (5.3) and the inequality (5.6), V1is a non-positive function and it satisfies the following differential equation: V 1(r)=−V1(r) r1+ ˙ϕ(u)2(r)(r+V1(r))2−1−¨ϕ(u)(r)(r+V1(r))2. (5.14) 3 Page 22 of 28 A. Martínez and A. L. Martínez-Triviño MJOM Take ε>0andRr0.Ifr≥Rand V1(r)≤−ε, from the sublinearity, we can suppose that −r/2<V1(r), and r2 4<(r+V1(r))2<(c+1) 2r2.(5.15) Now, choosing Rlarge enough, the Eq. (5.14) and the inequalities (5.7)and (5.15) give V 1(r)≥−1+ε r+rε 4˙ϕ(u)2(r)−(α+ε)(c+1) 2r.(5.16) Using the conditions (4.9) and the asymptotic behavior (5.13), Rmay be chosen large enough so that ˙ϕ(u)2(r)≥4 ε(α+ε)(c+1) 2r+1 rc+1−ε r,r≥R. Thus, if Ris large enough and r≥Rwhere V1(r)≤−ε, then V 1(r)≥c>0. Hence, V1(r)≥−εfor rlarge enough and we conclude the proof.  Lemma 5.6. limr→+∞1 r˙ϕ2(u)(r)V1(r)=−α. Proof. If λ(r)=1 r˙ϕ2(u)(r)V1(r), then from (5.3) and (5.12), we have λ(r)= ˙ϕ2(u)(r)2V1(r)¨ϕ(u)(r)1+V1(r) r−1 r2−1 r +˙ϕ2(u)(r)(r+V1(r))2 r(−¨ϕ(u)(r)−λ(r)). Fix ε>0andRlarge enough. Consider points r≥Rwhere λ(r)≥−α+, then −¨ϕ(u)(r)−λ(r)≤−¨ϕ(u)(r)+α−ε(5.17) and if Ris large enough, from (5.2) and (5.17), we also get that −¨ϕ(u)(r)−λ(r)≤−εα 2<0 and then λ(r)<−1 when Ris chosen sufficiently large. Hence, we obtain that λ(r)≤−α+εfor rlarge enough. In a similar way, we may prove that λ(r)≤−α−εfor rsufficiently large.  Now, (5.5) follows from (5.12), (5.13) and Lemmas 5.5 and 5.6. Theorem 5.7. (Case α=0) Assume that ˙ϕ(u0)r0≥u1,α=0and β> 0. Then, the problem (5.3)has an unique strictly convex C∞-solution uon [r0,∞[. Moreover, if lim u→+∞u¨ϕ(u)=0,(5.18) we have the following asymptotic expansion: u ˙ϕ(u)(r)=r−1 β2r+or−1,(5.19) MJOM Equilibrium of Surfaces in a Vertical Force Field Page 23 of 28 3 Proof. Arguing as in Theorems 4.5,4.11 and Proposition 4.12 ,(5.3)hasa unique C∞-solution uon [r0,∞[ which is strictly convex function satisfying that limr→∞ u(r)=∞. Moreover, as Lemmas 5.3 and 5.4 also work in this case, we have the following asymptotic expansion: u ˙ϕ(u)(r)=r+V1(r),(5.20) where V1verifies the same differential equation (5.14), is also non-positive and V1(r)→0. Moreover, from (5.2), ˙ϕwrites as ˙ϕ(u)(r)=β+o(1).(5.21) Consider now the new function V2(r)=r˙ϕ2(u)(r)V1(r). Then, V 2=r˙ϕ22¨ϕV1(r+V1)−1+(r+V1)2 r2(−r2¨ϕ−V2). From the expressions (5.18), (5.20)andL’Hˆopital’s rule, we have lim r→+∞¨ϕ(u(r)) r=0and lim r→+∞¨ϕ(u(r)) r2=0,(5.22) and working as in Lemma 5.6 we can prove that V2(r)→−1. Finally, the Theorem follows from the expansion (5.20)asr→+∞. 5.1. Proof of Theorem A If α>0, from (5.12) and (5.13), we can write log( ˙ϕ(u))(r)=αr2 2+Υ(r),(5.23) where Υ=(¨ϕ−α)r+¨ϕV1. Hence, as the first non-vanishing akis positive, for rlarge enough Υ is a decreasing function in rsuch that −∞ <c= limr→+∞Υ(r) otherwise from Lemma 5.6,(1.10), (5.23) and using L’Hˆopital’s rule,wehavethat +∞= lim r→+∞˙ϕ2(u)(r) = lim r→+∞ e2Υ e−αr2= lim r→+∞e2Υ e−αr2 =−lim r→+∞ ˙ϕ(u)2(r)((¨ϕ(u)(r)−α)r+¨ϕ(u)(r)V1(r)) αr =αa 1, which is a contradiction. Applying again L’Hˆopital’s rule to limr→+∞e2Υ−e2c e−αr2,wehave ˙ϕ2(u)(r)=eαr2+2c+O(1) and lim r→+∞O(1) = αa 1. Thus, from Lemma 5.6 and Theorem 5.2, ϕ(u)(r)=reαr2+2c+αa1r+o(r), and (1.11) follows by integration of the above expression. If α= 0 then, the condition (5.18) follows from (5.20) and we have that u ˙ϕ(u)(r)=r−1 β2r+or−1.(5.24) 3 Page 24 of 28 A. Martínez and A. L. Martínez-Triviño MJOM Now, by taking V3(r)=(V2(r)+1)r2,weget V 3=2V3 r+r3˙ϕ22¨ϕV1(r+V1)−1+(r+V1)2 r2−r2¨ϕ+1−V3 r2 =r˙ϕ22V3 ˙ϕ2r2+2r4¨ϕV1(r+V1) r2−r2+(r+V1)2 r2(−r4¨ϕ+r2−V3) =r˙ϕ2(r+V1)2 r2−r4¨ϕ+r21−r2 (r+V1)2−V3 +r˙ϕ22V3 ˙ϕ2r2+2r4¨ϕV1(r+V1) r2. But, from (5.20)andL’Hˆopital’s rule, we obtain lim r→+∞¨ϕ(u(r)) r4=−4a1 β2, lim r→+∞r21−r2 (r+V1)2=−2 β2 thus, by working as in Lemma 5.6, we prove that lim r→∞ V3(r)=−2+4a1 β2. Hence, u ˙ϕ(u)(r)=r−1 β2r−2−4a1 β4r3+or−3, and (1.12) follows from integration in the above expression. 6. Uniqueness of Globally Convex Solutions Along this section ϕ:]a, +∞[−→ Rwill be a regular function satisfying the expansion (1.10). For any θ∈[0,2π[, we consider v= (cos θ,sin θ,0) and denote by Πv(t)the vertical plane Πv(t)={p∈R3|p, v=t}(6.1) Definition 6.1. Let Σ1and Σ2be two arbitrary subsets of R3. We say that Σ1is on the right hand side of Σ2respect to Πv(t) and write Σ1≥vΣ2if and only if for every point q∈Πv(t) such that π−1(q)∩Σ1=∅and π−1(q)∩Σ2=∅, we have the following inequality: inf{p, v:p∈π−1(q)∩Σ1}≥sup{p, v:p∈π−1(q)∩Σ2}, where π:R3→Πv(t) denotes the orthogonal projection on Πv(t). MJOM Equilibrium of Surfaces in a Vertical Force Field Page 25 of 28 3 For an arbitrary subset Mof R3, we also consider the following subsets: Σ+(t):={p∈M:p, v≥t}. Σ−(t):={p∈M:p, v≤t}. Σ∗ +(t):={p+2(t−p, v)v∈R3:p∈Σ+(t)}. Σ∗ −(t):={p+2(t−p, v)v∈R3:p∈Σ−(t)}. From Theorem A, it is natural to study [ϕ, e3]-minimal surfaces whose behavior at infinity is of rotational type. To be more precise, Definition 6.2. We say that a [ϕ, e3]-minimal end Σ is smoothly asymptotic to a rotational-type example if Σ can be expressed outside a ball as a vertical graph of a function uΣso that, according to αis either positive or zero, one of the following expressions holds: ϕ(uΣ)(x)=Ce α|x|2+O|x|2,if α>0,(6.2) where Cis a positive constant or up to a constant, G(uΣ)(x)=|x|2 2−1 β2log(|x|)+O|x|−2,(6.3) if α=0andβ>0. Let Σ be an embedded [ϕ, e3]-minimal surface Σ with a single end smoothly asymptotic to a bowl-type example. Then, there exists R>0 large enough such that Σ ∩(R3\B(0,R)) is the vertical graph of a function uΣ verifying either (6.2)ifα>0or(6.3)ifα=0andβ>0. Lemma 6.3. There exists r1>Rsuch that if t>r 1then Σ+(t)is a graph over Πv(t). Proof. It is clear that when t>R,Σ +(t) has only one component which is unbounded. Moreover, if α>0 then from (6.2), ˙ϕ(uΣ)(x)(duΣ)x(v)≥2αe α|x|2x, vC+e−α|x|2g(|x|), where lim |x|→ g(|x|) |x|2=0. Hence, there exists r1large enough such that if x, v≥r1, then (duΣ)x(v)> 0, and in this case, the Lemma follows because Σ is embedded and Σ+(r1)∪ π(Σ+(r1)) bounds a domain in R3. When α= 0, a similar argument with (6.3) also works.  From Lemma 6.3,fixedt>r 1,Σ ∗ +(t)∩{p∈R3:p, e3>R}is the vertical graph of the function satisfying u∗ t(x)=uΣ(x+2(t−x, v)v) (6.4) Lemma 6.4. Consider a>0not depending on Rand 0>0.Then,forR large enough and t>a+x, v, we have u∗ t(x)−uΣ(x)> 0>0.