The Bernstein problem for elliptic Weingarten multigraphs

The Bernstein problem for elliptic Weingarten multigraphs Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Abstract We prove that any complete, uniformly elliptic Weingarten surface in Euclidean 3-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. 1. Introduction AWeingarten surface is an immersed surface Σin R3whose mean curvature Hand Gauss curvature Kare related by some smooth equation (1.1) W(H, K) = 0. For our purposes, it will suffice to require that Wis of class C2. We say that Σis an elliptic Weingarten surface if (1.1) is elliptic when viewed as a fully nonlinear second order PDE in local graphical coordinates on Σ. In the elliptic case, (1.1) can be rewritten as (1.2) H=g(H2−K),4t(g0(t))2<1for all t≥0, for some C2function g: [0,∞)→R; the inequality in (1.2) is precisely the ellipticity condition for the equation. Note that, when gis constant, (1.2) is the constant mean curvature (CMC) equation. Elliptic Weingarten surfaces are sometimes called special Weingarten surfaces. Their global geometry has been studied in depth by many authors; see e.g. [1, 2, 4, 6, 7, 9, 11, 12, 18, 19, 20, 24, 25, 26, 29]. The most fundamental open problem in the global theory of elliptic Weingarten surfaces is probably the Bernstein problem, see e.g. Rosenberg and Sa Earp [24]: Bernstein problem: Are planes the only entire elliptic Weingarten graphs in R3? If g(0) 6= 0, there are no entire graphs satisfying (1.2), as follows from an easy application of the maximum principle and the fact that spheres of radius 1/|g(0)|satisfy (1.2). That is, the Bernstein problem is only meaningful for Weingarten surfaces of minimal type, i.e., when g(0) = 0. The Bernstein problem has an affirmative answer when (1.2) is uniformly elliptic, that is, when there exists some constant Λ∈(0,1) such that (1.3) 4t(g0(t))2≤Λ<1for all t≥0. Mathematics Subject Classification: 53A10, 53C42, 35J15, 35J60. Keywords: Weingarten surfaces, fully nonlinear elliptic equations, Bernstein problem, multigraphs, curvature estimates, quasiconformal Gauss map. 2 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira This follows from a deep theorem by L. Simon ([27, Theorem 4.1]) about entire graphs with quasiconformal Gauss map, not necessarily satisfying a Weingarten equation; see e.g. [26]. Note that this result includes the classical Bernstein theorem for minimal surfaces (H= 0). Not much is known about classes of Weingarten surfaces for which the Bernstein problem can be solved, if (1.3) does not hold (see [23]). One of our contributions in this paper is to solve in the affirmative the Bernstein problem for a wide class of non-uniformly elliptic Weingarten equations; see the Corollary at the end of the introduction. The Bernstein problem is related to the spherical image of the Gauss map N: Σ →S2of elliptic Weingarten surfaces Σin R3. Indeed, note that if Σis a graph, N(Σ) lies in an open hemisphere. Conversely, if N(Σ) lies in some open hemisphere, then Σmight not be a graph, but it is a multigraph, i.e., a local graph with respect to a specific fixed direction of R3. A classical theorem by Hoffman, Osserman and Schoen [17] proves that if the Gauss map image N(Σ) of a complete CMC surface Σlies in a closed hemisphere, then Σis a plane (H= 0) or a cylinder (H6= 0). So, this theorem can be seen as a solution to a generalized Bernstein problem for CMC multigraphs, and motivates the following Bernstein problem for multigraphs: Are planes and cylinders the only complete, elliptic Weingarten surfaces in R3whose Gauss map image lies in a closed hemisphere of S2? Observe that this problem asks, in particular, if complete (not necessarily entire) elliptic Weingarten graphs in R3are planes. This time, in contrast with the case of entire graphs, the problem is non-trivial if g(0) 6= 0 in (1.2). Also, one should note that there exist complete, rotational CMC unduloids in R3whose Gauss map image lies in an arbitrarily small tubular neighborhood of a geodesic of S2. These examples show the necessity of the hypothesis on the Gauss map image in this problem. We now state the main results of this paper. In Section 2 we will show (see Lemma 2.2): Lemma A: If the Gauss map image N(Σ) of an elliptic Weingarten surface Σlies in a closed hemisphere, then either Σis a multigraph (i.e., N(Σ) lies in the interior of this hemisphere), or Σ is a piece of a plane or a cylinder. Thus, in order to classify elliptic Weingarten surfaces whose Gauss map image lies in a closed hemisphere, it suffices to classify elliptic Weingarten multigraphs. In Section 3 we will prove that the Bernstein problem for elliptic Weingarten multigraphs (and in particular for entire graphs) can be solved whenever we have a bound on the norm of the second fundamental form. From Lemma A and Theorem 3.1 we have: Theorem A: Planes and cylinders are the only complete elliptic Weingarten surfaces in R3with bounded second fundamental form and Gauss map image contained in a closed hemisphere. The proof of Theorem A is based on an argument by Hauswirth, Rosenberg and Spruck [16] in the context of CMC surfaces in the product space H2×R, where H2denotes the hyperbolic plane, and subsequent modifications of it in other geometric theories by Espinar and Rosenberg [10], and G´ alvez, Mira and Tassi [14]; see also Manzano-Rodr´ ıguez [21] and Daniel-Hauswirth [8]. Theorem A reduces the Bernstein problem for elliptic Weingarten graphs or multigraphs to the obtention of a priori estimates for the norm of the second fundamental form (usually called curvature estimates). In Section 4 we will prove such a curvature estimate for the uniformly elliptic Elliptic Weingarten surfaces 3 case (Theorem 4.2). This estimate is based on a paper by Rosenberg, Souam and Toubiana [28] on stable CMC surfaces in Riemannian 3-manifolds. A key difficulty in our situation is that, in the natural blow-up process that one uses to obtain such curvature estimates, the equation (1.2) is lost in the limit. That is, even if one finds a limit surface after the blow-up, this limit surface might not be, in general, an elliptic Weingarten surface anymore. As a consequence of this estimate, we obtain from Lemma A and Theorem 4.1: Theorem B: Planes and cylinders are the only complete, uniformly elliptic Weingarten surfaces in R3whose Gauss map image is contained in a closed hemisphere. Note that Theorem B extends the most general known result for the Bernstein problem (consequence of Simon’s theorem [27]) to the multigraph case, and also extends the Hoffman- Osserman-Schoen theorem from CMC surfaces to uniformly elliptic Weingarten surfaces. Thus, Theorem B solves the Bernstein problem for multigraphs in the uniformly elliptic case. In order to explain our results in the non-uniformly elliptic case, it is convenient to rewrite the elliptic Weingarten equation (1.2) in terms of the principal curvatures κ1, κ2of the surface as, (1.4) κ2=f(κ1). Here, fis a C2function on an open interval If⊂R, that satisfies f0<0(by ellipticity) and f◦f= Id (by symmetry of the relations 2H=κ1+κ2and K=κ1κ2). Moreover, Ifis of the form (a, ∞),(−∞, b)or R. The uniform ellipticity condition (1.3) is written for fin (1.4) as (1.5) 0<Λ1≤ −f0(x)≤Λ2for all x∈If=R, where Λ1,Λ2are positive constants. That is, the slope of the graph of fis negative and uniformly bounded away from 0and −∞. As a matter of fact, by the symmetry properties of f, it suffices to impose one of the two inequalities in (1.5) to obtain the other one. Note that If=Rfor any uniformly elliptic Weingarten equation. Thus, the most typical examples of non-uniformly elliptic Weingarten equations happen when the function fin (1.4) is not globally defined on R; for instance, this is the case of the linear Weingarten equation 2aH +bK =c, with a, b, c ∈Rsatisfying the ellipticity condition a2+bc > 0, and b6= 0. When If6=R, we use Theorem A and the family of parallel surfaces to give an affirmative answer to both the Bernstein problem and the generalized Bernstein problem for multigraphs. By Lemma A and Theorem 5.1 we have: Theorem C: Let Σbe a complete elliptic Weingarten surface in R3whose Gauss map image is contained in a closed hemisphere. Assume that the function fof its associated Weingarten relation (1.4) is not defined in all R. Then Σis a plane or a cylinder. Corollary: Planes are the only entire graphs in R3that satisfy an elliptic Weingarten equation (1.4) with If6=R. Bernstein’s problem remains open when If=Rand (1.2) is not uniformly elliptic. For an alternative formulation of the Corollary above when we write the Weingarten equation as (1.2) instead of as (1.4), see Theorem 5.3. 2. Elliptic Weingarten surfaces 2.1. The Weingarten equation. Let us start by clarifying some aspects about the different ways of writing an elliptic Weingarten equation. 4 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira First, a word of caution. It is not a good idea to work directly with the simple form (1.1) of the Weingarten equation, because it can be misleading. For instance, both planes and round spheres of radius 1/2satisfy the simple linear Weingarten equation K= 2H, which can be proved to be elliptic. At first sight, this would seem to contradict the maximum principle for elliptic PDEs. This is explained by the fact that the equation K= 2Hactually contains two different elliptic theories (see Figure 3.1 and the discussion below). The Weingarten equation (1.1) can be rewritten in terms of the principal curvatures κ1, κ2as (2.1) Φ(κ1, κ2) = 0, where Φ∈C2(R2)is symmetric, i.e. Φ(k1, k2) = Φ(k2, k1). With this formulation, the ellipticity condition for the Weingarten equation is written as (see e.g. [20], pg. 129) (2.2) ∂Φ ∂k1 ∂Φ ∂k2 >0if Φ=0. Thus, if (2.2) holds, we see using the symmetry of Φthat each connected component of Φ−1(0) ⊂R2can be written as a graph (2.3) k2=f(k1), where fis defined on an interval If:= (a, b)⊂R, and satisfies the following conditions: (i) fis C2, and f0<0(by ellipticity). (ii) f◦f= Id (by symmetry of Φ). (iii) If a6=−∞, then b= +∞and f(x)→+∞as x→a. (iv) If b6= +∞, then a=−∞and f(x)→ −∞as x→b. Each connected component of Φ−1(0) gives rise to a different elliptic theory, with different geometric properties. For instance, the already mentioned Weingarten relation K= 2Hcan be rewritten as Φ(κ1, κ2) = κ1+κ2−κ1κ2= 0, and it is clear that Φ−1(0) has two connected components; see Figure 3.1. In one of them, all surfaces have principal curvatures greater than 1, and so are convex, while all surfaces of the other connected component have non-positive curvature. Figure 2.1. The two connected components of Φ−1(0) in the (κ1, κ2)-plane, for the Weingarten equation Φ(κ1, κ2)=0corresponding to K= 2H. Elliptic Weingarten surfaces 5 Alternatively, and also by the symmetry and ellipticity conditions on Φ, it is easy to see that each connected component of Φ−1(0) can be seen as a graph of the form (2.4) k1+k2 2=g(k1−k2)2 4, where g∈C2([0,∞)) satisfies, by the ellipticity inequality (2.2), the condition (2.5) 4t(g0(t))2<1for all t≥0. This shows that there is no loss of generality in working with (1.2) or with (1.4) when dealing with a class of elliptic Weingarten surfaces in R3, and that both formulations are equivalent. Thus, a surface Σin R3is an elliptic Weingarten surface if its curvature diagram (κ1(Σ), κ2(Σ)) ⊂R2is a subset of a regular curve of R2of the form (2.3). We note that for graphs z=u(x, y), the Weingarten equation (1.2) is equivalent to the fully nonlinear elliptic PDE (2.6) F(ux, uy, uxx, uxy, uyy) = 0, where F(p, q, r, s, t) := H−g(H2−K)∈C2(R5), for (2.7) H(p, q, r, s, t) := (1 + q2)r−2pqs + (1 + p2)t 2(1 + p2+q2)3/2,K(p, q, r, s, t) := rt −s2 (1 + p2+q2)2. Here, the ellipticity of (2.6) is equivalent to the ellipticity condition (2.5) for g. In particular, if gverifies (2.5), the class Wgof elliptic Weingarten surfaces in R3given by (1.2) satisfies the maximum principle in its usual geometric version. The number α:= g(0) has a geometric meaning in the class of Weingarten surfaces Wg, and is called the umbilical constant of the class Wg, because umbilics of any surface in Wghave principal curvatures equal to α. Note that by making, if necessary, the change g(t)7→ −g(t) in (1.2) while reversing the orientation of the surface, we may assume without loss of generality that α=g(0) ≥0. If g(0) = 0, planes belong to the Weingarten class Wg, and the Weingarten equation (1.2) is said to be of minimal type. If g(0) >0, spheres of radius 1/α with their inner orientation are elements of Wg. When we write the Weingarten equation as (1.4), the umbilical constant αis given by the relation f(α) = α, and the equation is of minimal type if f(0) = 0. Moreover, if fis defined at x= 0 with f(0) 6= 0, the cylinders in R3of principal curvatures {0, f(0)}are elliptic Weingarten surfaces satisfying (1.4). 2.2. Weingarten multigraphs. Given an immersed oriented surface Σin R3with unit normal N: Σ →S2, we will refer to the set N(Σ) ⊂S2as the Gauss map image of Σ. Definition 2.1. A surface Σin R3is a multigraph if there is some plane Pin R3such that Σcan be seen locally around each point p∈Σas a graph over P. Equivalently, Σis a multigraph if its Gauss map image is contained in an open hemisphere of S2. After a change of Euclidean coordinates, we can always assume without loss of generality for a multigraph Σthat N(Σ) lies in the upper open hemisphere S2 +, and so, that ν > 0where ν:= hN, e3iis the angle function of Σ. Obviously, any graph z=u(x, y)is a multigraph. Note that the surfaces with vanishing angle function, ν≡0, are open pieces of flat surfaces of the form Γ×R, where Γis some immersed curve in R2. 6 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Lemma 2.2. Let Σbe an elliptic Weingarten surface, and assume that its angle function satisfies ν≥0. Then either ν≡0, or ν > 0on Σ. If ν≡0, then Σis a piece of a plane or a cylinder. Proof. Let fbe the smooth function defining the relation (1.4). If fis not defined at 0, then by properties (i)-(iv) of fit follows that Σhas positive curvature. In particular, its Gauss map N: Σ →S2is a local diffeomorphism, hence an open mapping. Thus, if ν≥0, it must actually happen that ν > 0, and Lemma 2.2 holds in this case. Assume next that fis defined at 0, and let q0∈Σsatisfy ν(q0)=0. Without loss of generality, we assume that q0is the origin and N(q0) = (1,0,0). If f(0) = 0 (resp. f(0) 6= 0), let Cdenote the vertical plane (resp. the vertical cylinder with principal curvatures 0and f(0)) that is tangent to Σat q0, with the same orientation. Note that both Σand Csatisfy the elliptic Weingarten relation (1.4). Thus, both Σand Ccan be seen around the origin as graphs x=hi(y, z),i= 1,2, over their common tangent plane, and h1, h2are solutions to the same C2fully nonlinear elliptic PDE, associated to (1.4). In these conditions, it is well known that the difference h=h1−h2 satisfies a second order, linear, homogeneous elliptic PDE L[h] = 0 with C1coefficients. Note that Dh(0,0) = (0,0), where Dh = (hy, hz). Also hz= (h1)z, since h2(y, z)does not actually depend on z, because it corresponds to the vertical plane (or cylinder) C. Assume that his not identically zero. Then, it is a standard fact (see e.g. Bers [3]) that there exist coordinates (u, v)obtained by a linear transformation of (y, z)such that hhas the local representation around the origin h(u, v) = w(u, v) + o(pu2+v2)k where w(u, v)is a homogeneous harmonic polynomial of degree k≥2. In particular, the image of Dh cannot lie in a half-plane around the origin. Thus, there exist points arbitrarily close to the origin where hz>0. Hence, ν < 0at those points, what contradicts that ν≥0in Σ. Therefore, hmust be identically zero, and so Σis a piece of the cylinder (or plane) C. This concludes the proof.  2.3. The linearized Weingarten equation. Let Σbe an immersed oriented surface in R3with unit normal N. Given φ∈C∞ 0(Σ), consider the normal variation of Σassociated to φ, (2.8) (p, τ)∈Σ×(−ε, ε)7→ p+τφ(p)N(p), and denote by H(τ)and K(τ)the mean curvature and Gauss curvature of the corresponding surface Στin (2.8). In [24] it is shown (see equation (1.1)) that (2.9) 2H0(0) = ∆φ+ (4H2−2K)φ, K0(0) = div(T1∇φ)+2HKφ. Here, H, K denote the mean curvature and Gauss curvature of Σ;∆,div,∇are the Laplacian, divergence and gradient operator on Σ, and T1:= 2HId −S, where Sis the shape operator of Σ. Assume now that Σsatisfies an elliptic Weingarten equation (1.2). Let {Στ}τ∈(−ε,ε)be a normal variation of Σassociated to some function φ∈C∞ 0(Σ), and denote (2.10) W(τ) := H(τ)−g(H(τ)2−K(τ)) : Σ ×(−ε, ε)→R, with the previous notation, where gis the function in (1.2). Then, taking into account (2.9), the linearized operator of the Weingarten equation (1.2) satisfied by Σis (2.11) W0(0) = Lg[φ] = 1−2gg0 2∆φ+g0div(T1∇φ) + qφ, Elliptic Weingarten surfaces 7 where g, g0are evaluated at H2−K, and q:= 2g2(1 −2gg0)−(1 −4gg0)K. We remark that Lgin (2.11) is a linear elliptic operator, since (1.2) is elliptic. 3. Weingarten multigraphs with bounded second fundamental form The present section is devoted to the proof of the following theorem: Theorem 3.1. Planes are the only complete elliptic Weingarten multigraphs with bounded second fundamental form. Proof. We will argue by contradiction. Let Σbe a complete elliptic Weingarten multigraph with bounded second fundamental form, and assume that Σis not a plane. Let f∈C2(If),If⊂R, be the function that defines the Weingarten relation (1.4) satisfied by Σ. We denote by α≥0the ellipticity constant of f, given by f(α) = α. Up to an Euclidean change of coordinates, we can assume that N(Σ) is a subset of S2 +. We will start by noting that fis defined at 0, i.e., that 0∈If. Indeed, otherwise we have from α≥0that If⊂(0,∞), and by the properties of fwe deduce that If= (a, ∞)for some a≥0, and f(x)→ ∞ (resp. f(x)→a) as x→a+(resp. x→ ∞). Thus, both principal curvatures of Σare positive, by monotonicity of f. Since Σhas bounded second fundamental form, we easily see from there that its Gaussian curvature satisfies K≥c > 0for some constant c. Thus, Σis compact, a contradiction with N(Σ) ⊂S2 +. So, from now on we will assume that there exists f(0).In the rest of the proof, Wfwill denote the class of all immersed oriented surfaces in R3that satisfy (1.4) for our choice of f. By ellipticity, surfaces in Wfsatisfy the maximum principle. Note that if f(0) 6= 0, then f(0) >0 (by monotonicity, since α > 0in this case). We proceed with the proof. Since N(Σ) ⊂S2 +, the surface Σcan be locally seen as a graph z=u(x, y). Take from now on an arbitrary point p∈Σ, and let R > 0be the largest value for which an open neighborhood V ⊂ Σof pcan be seen as a graph z=u(x, y)over D=D(bq, R), where bq=π(p), with π(x, y, z) := (x, y). See Figure 3.1. That this radius Rexists, i.e., that Ris not infinite, is proved in Assertion 3.2 below: Figure 3.1. Initial situation of the proof of Theorem 3.1. 8 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Assertion 3.2. In the previous conditions, R < ∞. Proof. If α > 0, any sphere Sαof radius 1/α lies in the Weingarten class Wffor its inner orientation. In case R > 1/α, we could place a sphere Sαabove the graph V, and then move it downwards until reaching a first contact at an interior point of both surfaces. This is a contradiction with the maximum principle for surfaces in Wf. Therefore, R≤1/α if α > 0. Assume now that α= 0, i.e., f(0) = 0, and so Σis a Weingarten surface of minimal type. Let κ1≥κ2denote the principal curvatures of Σ. Since |σ|2:= κ2 1+κ2 2is bounded, it is clear by monotonicity of fand the condition f(0) = 0 that the set (κ1(Σ), κ2(Σ)) ⊂R2lies on a wedge region Rof the form (3.1) R={(x, y) : x≥y, m1x≤y≤m2x}, m1, m2<0. This easily implies the inequality (3.2) κ2 1+κ2 2≤γ κ1κ2, for some γ∈Rat every point of Σ, a condition that is equivalent for any surface Σin R3to the property that its Gauss map N: Σ →S2is quasiconformal; see Section 16.6 in [15]. So, assume R=∞. Then, Σis an entire graph with quasiconformal Gauss map. By a theorem of L. Simon ([27, Theorem 4.1]), Σmust be a plane, a contradiction. This completes the proof of Assertion 3.2.  Remark 3.3. Recall that, by hypothesis, Σhas bounded second fundamental form, that is, we have |σ(p)| ≤ γ < ∞for some constant γ, for all p∈Σ. This implies a well-known uniformicity property for Σas a local graph, see e.g. Proposition 2.3 in [28]. In our conditions, this property implies that there exists some δ=δ(γ)>0for which the following holds: Any p∈Σhas a neighborhood Up⊂Σthat is a graph over the disk Bp(2δ)⊂TpΣcentered at the origin and of radius 2δof its tangent plane at p. Also, if udenotes the function that defines this graph in any such disk Bp(2δ), it holds |Du|<1in Bp(2δ), where Du denotes the Euclidean gradient of u. Moreover, there exists µ=µ(γ)>0, such that the C2norm of uis at most µ/2on any of the disks Bp(2δ); that is, ||u||C2(Bp(2δ)) < µ/2. We remark that δ, µ only depend on the bound γfor the second fundamental form of Σ, and not on por Σ.These numbers δ, µ > 0will be considered fixed from now on in the proof. Let us fix next some notation for the rest of the proof of Theorem 3.1. First, note that by Assertion 3.2 there exists some q∈∂D for which ucannot be extended to a neighborhood of q. We let C1:= Γ1×R,C2:= Γ2×R, denote the two vertical cylinders in R3of radius 1/f(0) that pass through q, and whose unit normals at qare orthogonal to ∂D; (if f(0) = 0,C1, C2are actually the same vertical plane with opposite orientations). Note that Ci∈ Wf,i= 1,2, for their inner orientation. We will let Γi(s)be an arclength parametrization of the circle (or straight line) Γi, with Γi(0) = q. Given s0∈R, ε > 0, for each i= 1,2we let Ni(s0, ε)denote the open one-sided tubular set (3.3) Ni(s0, ε) := {Γi(s) + τ ηi(s) : |s−s0|< δ, τ ∈(0, ε)} ⊂ R2, where ηi(s)denotes the unit normal of Γi(s)that, at q, points in the direction bq−q, where bqis the center of D. See Figure 3.2. Assertion 3.4. In the above conditions, there exists ε > 0such that u(x, y)extends smoothly to D∪Ni(0, ε), for some i= 1,2. Moreover, this extension satisfies that u(x, y)diverges to either Elliptic Weingarten surfaces 9 Figure 3.2. The two circles Γi,i= 1,2, that are tangent to ∂D at q, and their associated one-sided tubular domains Ni(0, ε)at q. +∞or −∞when (x, y)∈ Ni(0, ε)approaches Γi. Specifically, if f(0) 6= 0, we have that u→ ∞ (resp. u→ −∞) if Dlies in the interior (resp. exterior) of Γi. The proof of Assertion 3.4 essentially follows from some technical arguments developed by Hauswirth, Rosenberg and Spruck in [16] in the context of surfaces of constant mean curvature H= 1/2in the homogeneous product space H2×R. Thus, we will postpone the proof to the Appendix, where we will only give a sketch of it. Let us indicate that a key point of the proof in our case is the use of Nirenberg’s a priori C2,α-estimates for fully nonlinear elliptic equations in dimension two. Remark 3.5. We point out the following consequence of Assertion 3.4, for later use. Assume that we are in the conditions of Assertion 3.4 with f(0) 6= 0, call Γ := Γi, and suppose, for definiteness, that u→ −∞. Then, by the asymptotic behavior of u, and taking a smaller ε > 0 if necessary, the graph U0⊂Σgiven by z=u(x, y)on Ni(0, ε)can be seen as a normal graph over an open set of the limit cylinder Γ×R, of the form {(Γ(s), t) : |s|< δ, t ∈(−∞, t0(s))} ⊂ Γ×R, where t0(s)is a continuous function on [−δ, δ]. Moreover, U0lies in the exterior region of Γ×R, and converges asymptotically to Γ×Ras t→ −∞. Assertion 3.6. The statement of Theorem 3.1 holds if f(0) = 0. Proof. We first prove that if f(0) = 0, then Σis a graph. Note that since f(0) = 0, the cylinders Ci= Γi×Rare vertical planes, and the one-sided tubular domains Ni(0, ε)of Assertion 3.4 are open rectangles in R2. Using the notation of Assertion 3.4, take q0∈ Ni(0, ε)∩Dgiven by q0= Γi(δ/2) + τ0ηi(δ/2) for τ0∈(0, ε)small enough, and let p0= (q0, u(q0)) ∈Σ. Then, we can apply again the extension process in Assertion 3.4, but this time starting with p0instead of p= (bq, u(bq)). In this way, we see that ucan also be extended to a one-sided tubular domain Ni(δ/2, ε1), i.e., to (3.4) {Γi(s) + τ ηi(s) : s∈[−δ/2,3δ/2], τ ∈(0, ε1)} ⊂ R2 for some ε1>0, and in particular to the union Ni(0, ε)∪Ni(δ/2, ε1). By repeating this process, we conclude that ucan be extended to a union of domains Ni(kδ/2, εk)for k∈Z, and so, to a one-sided tubular neighborhood NΓof the straight line Γ⊂R2that is tangent to ∂D at q. Moreover u(x, y)→ ±∞as (x, y)→Γ. See Figure 3.3, left. We claim next that ucan actually be extended to the slab SΓof R2contained between Γand the line parallel to Γthat passes through bq. To see this, take for each θ∈(−π/2, π/2) the open 16 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira positive eigenvalues λ2 1, λ2 2given by (4.9) λ2 i=λ2 i(p, q) = 6 + p4+ 6q2+q4+p2(6 + 4q2)±pQ4(p2, q2) 8(1 + p2+q2)>0, where Q4(x, y)is the polynomial of degree 4 Q4(x, y) = x4+y3(8y−4) + 2x2y(14 + 9y)+(y2−2y−2)2+ 4x(2 + 10y+ 7y2+ 2y3). Moreover, it is easy to check from (4.9) that both λi(p, q)are bounded from below by a positive constant when we restrict to the compact set Θ. So, after an orthogonal change of coordinates (r, s, t)7→ (¯r, ¯s, ¯ t), where the related orthogonal matrix depends on (p, q), we can write (H2−K)(p, q, r, s, t) = λ2 1¯r2+λ2 2¯ t2, where here ¯r, ¯ tdepend on (p, q, r, s, t), the dependence on (r, s, t)being linear. All these functions λi,¯r, ¯s, ¯ tcan be chosen to be real analytic in their arguments, except around the points (p, q, r, s, t)where λ1(p, q) = λ2(p, q), i.e., around the points where the eigenvalue multiplicity changes. Call Bto this set of points. We claim that B ∩ Θis empty. To see this, first observe that, by (4.9), Bis given by the expression Q4(p2, q2)=0. We can rewrite Q4as Q4(x, y) = ((x+y)2−2(x+y)−2)2+ 4xy(10 + x2+ 10y+y2+x(10 + 3y)). So, Q4(p2, q2)≥(p2+q2)2−2(p2+q2)−22, and the expression in the right hand side vanishes only when p2+q2= 1 + √3. By the definition of Θin (4.7), it is clear then that Q4(p2, q2)>0in Θ, since p2+q2≤9/4in Θ. Thus, Θdoes not intersect B. In particular, the functions λi,¯r, ¯s, ¯ tare real analytic in Θ. Now, note that for any w∈ {p, q, r, s, t}we have in Θ (4.10)  (H2−K)w √H2−K =|(λ2 1)w¯r2+ (λ2 2)w¯ t2+ 2λ2 1¯r¯rw+ 2λ2 2¯ t¯ tw| pλ2 1¯r2+λ2 2¯ t2 ≤|(λ2 1)w¯r| λ1 +|(λ2 2)w¯ t| λ2 +|2λ1¯rw|+|2λ2¯ tw| ≤C1=C1(Θ) for some positive constant C1depending on Θ. From here and (4.6), we have in Θ: (4.11) |Fn w|=Hw−√H2−KG0 n(H2−K)(H2−K)w √H2−K , ≤maxΘ|Hw|+1 2C1(Θ) ≤C2(Θ) where we have used that √t|G0 n(t)|<1/2by the ellipticity condition on G(t). Thus, all the first derivatives of Fnwith respect to any w∈ {p, q, r, s, t}are uniformly bounded in Θ. This proves property (a). Once we know that (a) holds, the proof of (b) is a straightforward consequence of the fact that all the equations (4.5) are uniformly elliptic on Θfor the same ellipticity constant, since all the Elliptic Weingarten surfaces 17 Gnsatisfy the uniform condition 4t(G0 n(t))2≤Λ<1for the same Λ. Thus, (4.8) holds for some λ=λ(Λ,Θ). With this, we are in the conditions of Theorem I in [22] (alternatively, see also Theorem 17.9 in [15]), which implies what follows in our situation. Fix δ0∈(0, δ0)once and for all, from now on. Then, there exist constants C0>0and α∈(0,1) such that (4.12) ||vn||C2,α(Bδ0)≤C0 for all n. Here C0, α depend only on Λ, in the following sense: at first, these constants depend on δ0, µ0, δ0, the ellipticity constant Λand the bounds on the derivatives of Fnin Θ. Nonetheless, δ0, µ0are determined by the condition that |bσn| ≤ 2, and the bounds (4.11) obtained for Fn won Θare independent of the equation Fn, i.e., they only depend on δ0. Since δ0has been considered fixed, the numbers C0, α only depend on Λ. Step 3: Existence and properties of a limit surface of the blown-up immersions It follows by the estimate (4.12) that the set {vn}nis bounded in the C2,α(Bδ0)-norm, and therefore is precompact in the C2,β(Bδ0)-norm, for any β∈(0, α). Thus, by the Arzela-Ascoli theorem, a subsequence of the vnconverges uniformly in the C2,β(Bδ0)-norm to some function v0∈C2,β(Bδ0); here, βis any number in (0, α), that we also consider fixed from now on. Once here, we can apply a typical diagonal extension process and deduce that the graph x3=v0(x1, x2)can be extended to a complete immersion ψ0: Σ0→R3that, by construction, is a limit in the C2-topology on compact sets of a subsequence of the immersions ϕn. We denote this limit surface simply by Σ0. That Σ0is complete follows since the radii of the b Dngo to ∞. Note that Σ0is not, in general, an elliptic Weingarten surface, since as explained before, the elliptic Weingarten equations (4.4) do not necessarily converge C1to a Weingarten equation. We single out the following list of properties of Σ0, that will be proved below. (P1) Σ0is complete. (P2) Σ0has bounded second fundamental form. (P3) The Gauss map image N(Σ0)lies in the closed hemisphere S2 +. (P4) The Gauss map N: Σ0→S2is quasiconformal. (P5) Σ0has a local uniform C2,β-estimate, in the following sense: there exist δ0, C0, β such that Σ0can be locally seen around each p∈Σ0as a graph over the disk D(0, δ0)⊂TpΣ0, and so that the C2,β-norm in D(0, δ0)of the corresponding graphing function is at most C0. (P6) Σ0is not a plane. The fact that Σ0is a complete surface was explained above. The second fundamental form of Σ0is bounded since it is a C2-limit of the immersions ϕn(b Dn), and |bσn| ≤ 2on b Dn. That N(Σ0)lies in S2 +is also immediate, since all the ϕnare multigraphs (here S2 +denotes the upper hemisphere in the original (x, y, z)-coordinates of R3). The local uniform C2,β-estimate for Σ0in (P5) follows directly from the previous arguments using Nirenberg’s theorem. Since the norm of the second fundamental form of ϕn(b Dn)is equal to 1at the origin for all n, the same happens to Σ0; thus, Σ0is not a plane. So, the only property that remains to check is (P4), i.e., that Σ0has quasiconformal Gauss map. To start, let us rewrite the uniformly elliptic Weingarten equation (4.4) satisfied by ϕnin the form (1.4); that is, we rewrite (4.4) as κ2=fn(κ1),where fn∈C2(R)satisfies fn◦fn= Id and the 18 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Figure 4.1. The region Rn. uniform ellipticity condition (1.5). Let κn 1≥κn 2denote the principal curvatures of ϕn. Then, by the bounds in (1.5), it is clear that there exist m1, m2<0(independent of n) such that, for each n, the set (κn 1(b Dn), κn 2(b Dn)) ⊂R2 lies in the wedge region of the plane Rn:= {(x, y) : x≥y, m1(x−αn)≤y−αn≤m2(x−αn)} ⊂ R2, where αnis the umbilical constant of (4.4), given by Gn(0) = αn, or equivalently by fn(αn) = αn. See Figure 4.1. Note that αn=g(0)/λn→0as n→ ∞. Thus, the regions Rnconverge to the region Rin (3.1), and it follows that the (bounded) set (κ1(Σ0), κ2(Σ0)) ⊂R2lies inside this wedge region R, where κ1≥κ2are the principal curvatures of Σ0. By the arguments explained after (3.1), we deduce that the Gauss map of Σ0is quasiconformal, as claimed. Step 4: A surface Σ0with the properties (P1)-(P6) of Step 3 cannot exist. To start, let us recall that in Section 3 we proved that if Σsatisfies properties (P1)-(P3), and additionally Σis an elliptic Weingarten surface of minimal type (i.e., f(0) = 0 in (1.4)), then Σmust be a plane; see Assertion 3.6. However, by inspecting this proof (including the proof of Assertion 3.4 in the Appendix), we can realize that the Weingarten condition in the case f(0) = 0 is used quite mildly. Specifically, it is only used to prove the following facts (see Remarks 3.7 and 6.1 for this matter): i) That, because |σ|is bounded, we have by Nirenberg’s theorem a uniform local C2,βestimate on Σin the sense of (P5) above. In particular, we can take limits (up to subsequence) of translations of Σin the C2norm on compact sets. ii) That Lemma 2.2 holds for any of the limit surfaces of Σby translations. iii) That Σhas quasiconformal Gauss map. This indicates that we can reproduce the proof of Assertion 3.6 in our context, provided that we can check first that the properties i), ii), iii) above hold for the limit surface Σ0. But this is clear in our situation. Indeed, since N: Σ0→S2is quasiconformal by (P4) and we have uniform local C2,β-estimates on Σ0by (P5), we only need to check that Lemma 2.2 holds for Σ0and for its limits by translations. That it holds for Σ0is immediate, since quasiconformal maps are open. Elliptic Weingarten surfaces 19 In addition, note that Σ0satisfies the quasiconformal equation (3.2) for some constant γ. Thus, any of its limit surfaces also satisfies (3.2), what implies that its Gauss map is also quasiconformal (and hence, open). So, Lemma 2.2 holds for these limit surfaces. Let us point out that, for the case of surfaces with quasiconformal Gauss map, the situation ν≡0in Lemma 2.2 implies that the surface is a plane, since cylinders do not have quasiconformal Gauss map. Consequently, we can deduce that a surface Σ0satisfying properties (P1)-(P5) of Step 3 must be a plane. But since Σ0is not a plane in our situation by (P6), this proves Step 4. This gives the desired contradiction, and completes the proof of Theorem 4.2.  5. A Bernstein-type theorem in the non-uniformly elliptic case For this section, we recall that if we write an elliptic Weingarten equation as (1.4), i.e., as κ2=f(κ1), the notation Ifindicates the domain of the function f, which is an interval of R. We stress that Theorem 5.1 below is new even for the case of entire graphs. Theorem 5.1. Let Σbe a complete multigraph in R3that satisfies an elliptic Weingarten equation κ2=f(κ1), with If6=R. Then Σis a plane. Proof. Since If6=R, we may take t06∈ If. It follows then by the symmetry property f◦f= Id of fthat there exists ε > 0such that (5.1) |κi(p)−t0| ≥ ε, ∀p∈Σ, i = 1,2; here κ1, κ2are the principal curvatures of Σ. We may take t06= 0 without loss of generality. Write ψ: Σ →R3for the immersion of Σinto R3, and consider for a:= 1/t0the parallel surface of Σat a distance a, given by ψa:= ψ+aN : Σ →R3, where N: Σ →S2is the Gauss map of Σ. In general, ψamay have singular points; indeed, its induced metric gacan be expressed at any point in terms of an orthonormal basis of principal directions {e1, e2}of Σas ga(ei, ej) = (1 −aκi)2δij, where κiis the principal curvature of Σin the direction ei. However, in our present situation, the condition (5.1) ensures that ψais everywhere regular. Moreover, it also follows from (5.1) and the expression of gathat ga(ei, ej)≥a2ε2δij, and so ψais a complete surface. In addition, the Gauss map of ψais equal to N(thus ψais also a multigraph), and {e1, e2}are also principal directions for ψa. The principal curvatures of ψaare given then by (5.2) κa i=κi 1−aκi , i = 1,2. From this expression and (5.1), it is clear that the κa iare uniformly bounded, i.e., that ψahas bounded second fundamental form. Finally, observe that the surface ψais also an elliptic Weingarten surface, since from κ2= f(κ1)and (5.2) it holds κa 2=fa(κa 1)for the function fa(t) := f(t 1+at ) 1−af(t 1+at ). Note that, by a simple computation, fa◦fa= Id and (fa)0<0, so this Weingarten equation is certainly elliptic. To sum up, ψais a complete elliptic Weingarten multigraph with bounded second fundamental form. By Theorem 3.1, it is a plane. Hence, Σmust also be a plane.  20 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira It is interesting to point out that the condition that If6=Rwas only used above to ensure the existence of a value t0∈Rfor which (5.1) holds. Thus, essentially the same proof above gives the following result, where we allow that If=R: Corollary 5.2. Let Σbe a complete elliptic Weingarten multigraph whose principal curvatures κ1, κ2satisfy |κi(p)−t0| ≥ ε, ∀p∈Σ, i = 1,2, for some t0∈Rand some ε > 0. Then Σis a plane. For the sake of completeness, we reformulate Theorem 5.1 for the situation in which the Weingarten equation is written as (1.2), instead of (1.4): Theorem 5.3. Let g∈C2([0,∞)) satisfy: (1) 4t(g0(t))2<1for all t(ellipticity condition). (2) Either t+g(t2)or t−g(t2)is bounded in [0,∞). Then, any complete multigraph (in particular, any entire graph) in R3that satisfies the Weingarten equation H=g(H2−K)is a plane. Proof. By Theorem 5.1, we only need to check that the second condition on gin the statement is equivalent to the fact that If6=R, for the function fappearing when we rewrite (1.2) as (1.4). Denote t:= H2−K, and note that {κ1, κ2}=g(t)±√t, because of (1.2). By the symmetry and monotonicity properties of f, the fact that fis not globally defined in Ris equivalent to the fact that one of {κ1, κ2}is globally bounded from above or from below on the graph κ2=f(κ1). This easily gives the equivalence of If6=Rwith the second condition above.  Conditions (1)-(2) in Theorem 5.3 have also appeared in previous works by Sa Earp and Toubiana [25, 26] in connection with the existence of catenoids and half-space theorems for elliptic Weingarten surfaces of minimal type. See also [9]. An elliptic linear Weingarten surface is one that satisfies the equation (5.3) 2aH +bK =c, a, b, c ∈R, where the ellipticity condition is a2+bc > 0. This family contains surfaces of constant mean curvature (b= 0) and of constant positive curvature (a= 0), and corresponds to the family of parallel surfaces of the class of CMC surfaces in R3. However, as the parallel surface procedure usually creates singularities, their global geometry is not equivalent to the class of CMC surfaces. In terms of κ1, κ2, equation (5.3) is written as κ2=f(κ1), where f(x) = c−ax a+bx, which is not globally defined on Runless b= 0. Thus, we have: Corollary 5.4. Planes and cylinders are the only complete, elliptic linear Weingarten surfaces in R3whose Gauss map image lies in a closed hemisphere of S2. Proof. If b= 0, this is the classical theorem of Hoffman, Osserman and Schoen for CMC surfaces, see [17]; note that it also follows from Theorem 4.1. Also, if b6= 0, since fis not globally defined in R, it follows from Theorem 5.1 that planes are the only complete multigraphs that satisfy (5.3). Finally, by Lemma 2.2, we see that if the surface is not a multigraph, it must be a cylinder. This completes the proof.  Elliptic Weingarten surfaces 21 6. Appendix: proof of Assertion 3.4 Assume that we are in the conditions of Assertion 3.4, with the notations explained just before its statement. We give only a sketch of the proof, which is essentially identical to the original one by Hauswirth-Rosenberg-Spruck in [16] for CMC surfaces in H2×R. Let {qn}n⊂Dconverge to q∈∂D, and denote pn:= (qn, u(qn)) ∈ V ⊂ Σ. If ν(≥0) denotes the angle function of Σ, then we have ν(qn)→0, since ucannot be extended as a graph around q(one should have in mind here Remark 3.3). Therefore, the tangent planes TpnVbecome vertical as n→ ∞. Up to a subsequence, assume that {N(pn)}nconverges to some N0∈∂S2 +. Denote Vn:= Φn(V), where Φnis the vertical translation of R3sending pnto (qn,0). Note that all Vnhave uniformly bounded second fundamental form, and satisfy the Weingarten equation (1.4). Then, by Nirenberg’s a priori C2,α-estimates for fully nonlinear elliptic equations in dimension two (see [22, Theorem I]), we can use a standard compactness argument using the Arzela-Ascoli theorem and prove that the surfaces Vnconverge (up to subsequence) in the C2- norm in compact sets to some limit surface that also satisfies (1.4). Since ν(qn)→0and ν≥0, it follows from Lemma 2.2 that this limit surface is a piece of the cylinder Γ×R, where Γis the circle of radius 1/f(0) that passes through qwith interior unit normal N0. (If f(0) = 0,Γis an oriented straight line with the same properties). The circle Γmust be tangent to ∂D at q. Indeed, if Γand ∂D were transversal at q, there would be points in Γ∩Dnear qaround which the function uis well defined, but its gradient blows up when we approach the point (since the associated tangent planes become vertical), and this is not possible. Thus the cylinder Γ×Ris one of the cylinders Γi×Rdefined before Assertion 3.4. Take now a small segment γcontained in D, that ends at q. Since the tangent planes of V become vertical as we approach qthrough γ, it is clear that the restriction of uto γis monotonic, for values sufficiently close to q. Thus, it has a limit, which cannot be a finite number, by completeness of Σ. In other words, the restriction of uto any such segment γdiverges to +∞or to −∞. As a matter of fact, since Σwas oriented so that its angle function is positive it is easy to see, when f(0) 6= 0, that u→ ∞(resp. u→ −∞) if and only if the interior unit normal N0to Γ=Γi at qpoints in the direction q−bq(resp. bq−q). For each p∈ V, let Up⊂ V denote the neighborhood of pthat can be seen as a normal graph over D(0,2δ)⊂TpΣ, where δ > 0is the one in Remark 3.3. Consider the normal segment in R2 given by γ(t) := q±tN0, where the sign is chosen so that γ(t)lies in Dfor 0< t < t0with t0 small enough. Define the open set Σt0=[ 0<t<t0U(γ(t),u(γ(t))) ⊂Σ, which is a connected neighborhood of the curve {(γ(t), u(γ(t))) : 0 <t<t0} ⊂ Σ.By the convergence properties already proved, the projection of Σt0into R2contains a one-sided tubular domain Ni(0, ε0)as in (3.3), for the i∈ {1,2}such that Γ=Γi. See Figure 6.1. For each s∈[−δ, δ], denote by P(s)the vertical plane that is normal to Γat the point Γ(s). For t0small enough, P(s)intersects Σt0transversely for all s∈[−δ, δ]. Note that all points in Σt0∩P(0) belong to the curve (γ(t), u(γ(t))), and so Σt0∩P(0) is a connected graphical curve that does not intersect the cylinder Γ×R. In the same way, by transversality and the definition of Σt0, there is some t0>0and some ε > 0such that for each s∈[−δ, δ],Σt0∩P(s)is exactly 22 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Figure 6.1. The domain Σt0in Σ, and its projection, in the case f(0) = 0. one curve, which is a graph over a segment in R2of the form Γ(s) + tη(s). Here, η(s)is the unit normal of Γ(s)in (3.3) and tvaries in an interval Isthat contains (0, ε). All these properties let us conclude that Σt0is a graph when we restrict to the points of Σt0that project onto the one-sided tubular domain Ni(0, ε), for the value ε > 0above. Thus, ucan be extended as a graph to D∪Ni(0, ε), for some i∈ {1,2}. Let us also point out that (for t0>0small enough) Σt0does not intersect Γ×R. Indeed, otherwise there would exist a smallest (in absolute value) s1such that Σt0∩P(s1)intersects Γ×R. But Σt0∩P(0) does not intersect Γ×R, as explained above, so s1>0. By continuity we would have that Σt0∩P(s1)intersects Γ×Rbut it does not cross it. Hence, there would exist a point in Σt0∩P(s1)where the tangent plane to Σis vertical, and this is not possible since Σis a multigraph. The fact that Σt0does not intersect Γ×Rtogether with the previously proved asymptotic convergence of the curves Σt0∩P(s)to Γ×Rgive the asymptotic behavior of the statement of Assertion 3.4. This completes the (sketch of) proof of Assertion 3.4. Remark 6.1. For the purposes of the proof of Theorem 4.2, it is important to observe that the hypothesis that Σis an elliptic Weingarten surface has been used very mildly in this proof of Assertion 3.4. Indeed, it has only been used for the following two purposes: (1) To obtain the a priori local C2,α-estimate on compact sets given by Nirenberg’s theorem. By this estimate, we can ensure convergence up to subsequence in the C2topology on compact sets of limits of translations of Σin R3. (2) To ensure that any of the limit surfaces of Σobtained by translations via the previous C2,α-estimate satisfies the statement of Lemma 2.2. In particular, these limits are either multigraphs, or vertical cylinders or planes. References [1] J.A. Aledo, J.M. Espinar, J.A. G´ alvez, The Codazzi equation for surfaces, Adv. Math. 224 (2010), 2511–2530. [2] A.D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad Univ. 11 (1956), 5–17. (English translation: Amer. Math. Soc. Transl. 21 (1962), 341–354). Elliptic Weingarten surfaces 23 [3] L. Bers, Remark on an application of pseudoanalytic functions, Amer. J. Math. 78 (1956), 486–496. [4] R. Bryant, Complex analysis and a class of Weingarten surfaces. preprint. arXiv:1105.5589. [5] A. Bueno, J.A. G´ alvez, P. Mira, The global geometry of surfaces with prescribed mean curvature in R3,Trans. Amer. Math. Soc. 373 (2020), 4437–4467. [6] S.S. Chern, On special W-surfaces. Proc. Amer. Math. Soc. 6(1955), 783–786. [7] A.V. Corro, W. Ferreira, K. Tenenblat, Ribaucour transformations for constant mean curvature and linear Weingarten surfaces, Pacific J. Math. 212 (2003), 265–297. [8] B. Daniel, L. Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. (3),98 no.2 (2009), 445–470. [9] J.M. Espinar, H. Mesa, Elliptic special Weingarten surfaces of minimal type in R3of finite total curvature, preprint (2019). arXiv:1907.09122 [10] J.M. Espinar, H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in M2×R. J. Diff. Geom. 82 (2009), 611–628. [11] J.A. G´ alvez, A. Mart´ ınez, F. Mil´ an, Linear Weingarten surfaces in R3,Monatsh. Math 138 (2003), 133–144. [12] J.A. G´ alvez, P. Mira, Uniqueness of immersed spheres in three-manifolds, J. Diff. Geom., to appear. [13] J.A. G´ alvez, P. Mira, Rotational symmetry of Weingarten spheres in homogeneous three-manifolds, preprint (2018), arXiv:1807.09654. [14] J.A. G´ alvez, P. Mira, M.P. Tassi, Complete surfaces of constant anisotropic mean curvature, preprint, (2019), arXiv:1912.0194 . [15] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, (Classics in mathematics) Springer (2001). [16] L. Hauswirth, H. Rosenberg, J. Spruck, On complete mean curvature 1 2surfaces in H2×R,Comm. Anal. Geom. 5(2008) 989–1005. [17] D. Hoffman, R. Osserman, R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in R3 and R4,Comment. Math. Helv. 57 (1982), 519–531. [18] P. Hartman, A. Wintner, Umbilical points and W-surfaces, Amer. J. Math. 76 (1954), 502–508. [19] H. Hopf, Uber Flachen mit einer Relation zwischen den Hauptkrummungen, Math. Nachr. 4(1951), 232–249. [20] H. Hopf. Differential Geometry in the Large, volume 1000 of Lecture Notes in Math. Springer-Verlag, 1989. [21] J.M. Manzano, M. Rodr´ ıguez, On complete constant mean curvature vertical multigraphs in E(κ, τ),J. Geom. Anal. 25 (2015), 336–346. [22] L. Nirenberg, On Nonlinear Elliptic Partial Differential Equations and Holder Continuity, Comm. Pure App. Math. 6(1953), 103–156. [23] Th.M.Rassias, R.Sa Earp, Some problems in analysis and geometry. Complex Analysis in Several Variables, Hadronic Press, Florida 1999, pp. 111–122. [24] H. Rosenberg, R. Sa Earp, The geometry of properly embedded special surfaces in R3; e.g. surfaces satisfying aH +bK = 1, where aand bare positive, Duke Math. J. 73 (1994), 291–306. [25] R. Sa Earp, E. Toubiana, Classification des surfaces de type Delaunay, Amer. J. Math. 121 (1999), 671–700. [26] R. Sa Earp, E. Toubiana, Sur les surfaces de Weingarten sp´ eciales de type minimal, Bull. Braz. Math. Soc. 26 (1995), 129–148. [27] L. Simon, A H¨ older estimate for quasiconformal maps between surfaces in Euclidean space, Acta Math. 139 (1977), 19–51. [28] H. Rosenberg, R. Souam, E. Toubiana, General curvature estimates for stable H-surfaces in 3-manifolds and applications. J. Diff. Geom. 84 (2010), 623–648. [29] K. Tenenblat, Transformations on manifolds and applications to differential equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, 1998. Isabel Fern´ andez Departamento de Matem´ atica Aplicada I, Instituto de Matem´ aticas IMUS Universidad de Sevilla (Spain). e-mail: [email protected] Jos´ e A. G´ alvez 24 Isabel Fern´ andez, Jos´ e A. G´ alvez, Pablo Mira Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Granada (Spain). e-mail: [email protected] Pablo Mira Departamento de Matem´ atica Aplicada y Estad´ ıstica, Universidad Polit´ ecnica de Cartagena (Spain). e-mail: [email protected] Research partially supported by MINECO/FEDER Grant no. MTM2016-80313-P