ON SURFACES OF PRESCRIBED F -MEAN CURVATURE

Ulrich Clarenz and Heiko von der Mosel

Hypersurfaces of prescribed weighted mean curvature, or F -mean curvature, are introduced as critical immersions of anisotropic surface energies, thus generalizing minimal surfaces and surfaces of prescribed mean curvature. We first prove enclosure theorems in ℝ^{n+1} for such surfaces in cylindrical boundary configurations. Then we derive a general second variation formula for the anisotropic surface energies generalizing corresponding formulas of do Carmo for minimal surfaces, and Sauvigny for prescribed mean curvature surfaces. Finally we prove that stable surfaces of prescribed F -mean curvature in ℝ^{3} can be represented as graphs over a planar strictly convex domain Ω, if the given boundary contour in ℝ^{3} is a graph over ∂Ω.

1. Introduction and main results

Let X : M → ℝ^{n+1}, n ≥ 2, be an immersion of class C^{3}(M, ℝ^{n+1}) of an n-dimensional smooth manifold M = M^{n} with boundary ∂M into ℝ^{n+1}. We denote the corresponding unit normal by N and the induced area element by dA, and consider general parametric variational functionals F of the form

F(X) := ∫_{M} F (X, N) dA. (1.1)

The integrand F of class C^{0}(ℝ^{n+1} × ℝ^{n+1}) ∩ C^{3}(ℝ^{n+1} × (ℝ^{n+1}\{0})) is a parametric Lagrangian characterized by the homogeneity condition

F (y, tz) = tF (y, z) for all t > 0, (y, z) ∈ ℝ^{n+1} × ℝ^{n+1}. (H)

Note that (H) implies

F_{zz}(y, z)z = 0 for all (y, z) ∈ ℝ^{n+1} × (ℝ^{n+1}\{0}); (1.2)

hence we will identify the symmetric endomorphism F_{zz}(y, z) : ℝ^{n+1} → ℝ^{n+1} with its restriction to the space

z^{⊥} := {ζ ∈ ℝ^{n+1} : ⟨ζ, z⟩ = 0}. (1.3)

Important examples of parametric Lagrangians are given by the area integrand

A(z) := |z|, (1.4) and the integrand

E(y, z) := |z| + ⟨Q(y), z⟩ (1.5)

appearing in the theory of capillary surfaces. Here, Q can be chosen as a differentiable vectorfield in ℝ^{n+1} with div_{ℝ^{n+1}}Q(y) = H(y), where H(y) is a given function representing the prescribed mean curvature. Critical immersions of the corresponding functionals

A(X) := ∫_{M} A(N) dA = ∫_{M} dA (1.6)

and

E(X) := ∫_{M} E(X, N) dA (1.7)

are minimal surfaces and surfaces of prescribed mean curvature H(X), respectively.

Another interesting example is

F (z) = ∑_{j=1}^{3} \sqrt{δ^{2}|z|^{2} + z_{j}^{2}}, δ > 0, (1.8)

which serves as a regularized version of the discrete l^{1}-norm used for numerical computations involving the anisotropic mean curvature flow [7]. Furthermore, in surface processing [3] such parametric functionals have become an increasingly important tool to enhance edge structures within a suitable surface evolution based on (1.6) and (1.8) . For more examples of integrands and applications in numerical analysis we refer to [8] and [6].

For general parametric integrals we recall the notion of the F -mean curvature

H_{F} (X, N) = H_{F} := − tr (A_{F} S), (1.9)

as introduced in [2, 4]. Here, S ∈ End (T M) is the shape operator defined by DX ∘ S := DN on the tangent bundle T M, and A_{F} ∈ End (T M) is the symmetric endomorphism field given by

A_{F} := (DX)^{−1}(F_{zz}(X, N)DX) on T M. (1.10)

For the special parametric Lagrangians in (1.6) and (1.7) the F -mean curvature H_{F} reduces to the classical mean curvature H, since A_{F} |_{T_{w}M} = Id_{T_{w}M} for each w ∈ M and F (y, z) = A(z), or F (y, z) = E(y, z), respectively. Here T_{w}M denotes the tangent space of M at w ∈ M.

The first author proved in [2] that the Euler equation for F can be written as

H_{F} = ∑_{i=1}^{n+1} F_{y^{i}z^{i}}(X, N). (1.11)

Consequently, given a general parametric Lagrangian F = F (y, z), critical immersions of the corresponding parametric functional F may be viewed as surfaces of prescribed F -mean curvature. In particular, we will regard critical immersions of the specific parametric functional

F^{0}(X) := ∫_{M} F (N) dA + ∫_{M}⟨Q(X), N⟩ dA, (1.12)

where div_{ℝ^{n+1}}Q(y) = H_{F} (y) ∈ C^{0}(ℝ^{n+1}) is a given function, as surfaces of prescribed F -mean curvature H_{F} (X). This class of surfaces yields a natural generalization of minimal surfaces if H_{F} (y) ≡ 0, or of surfaces of constant mean curvature if H_{F} (y) ≡ H_{F}^{0} ∈ ℝ. Let us point out that the parametric Lagrangian F (z) in (1.12) depends on z only, and that in case H_{F} (y) ≡ H_{F}^{0} ∈ ℝ the second integrand in (1.12) is linear in y and z and can be interpreted as a volume term.

As a starting point for our investigations we will derive in Section 2 a differential equation for the surface normal of an arbitrary immersion in terms of the F -Laplace-Beltrami operator of X

(1.13) Δ_{F} := div (A_{F} grad (.) ),

where the differential operators are taken with respect to the induced metric

g_{w}(V, W) = g(V, W) := ⟨DX(V), DX(W)⟩ for V, W ∈ T_{w}M, w ∈ M, (1.14)

i.e., div = div_{M} and grad = grad_{M}.

Theorem 1.1. Let N be the normal of an arbitrary immersion X of class C^{3}(M, ℝ^{n+1}) and let F ∈ C^{0}(ℝ^{n+1} × ℝ^{n+1}) ∩ C^{3}(ℝ^{n+1} × (ℝ^{n+1}\{0})) be a parametric Lagrangian. Then

Δ_{F} N + tr (A_{F} S^{2})N = DX(div (SA_{F})) . (1.15)

Here, div (SA_{F}) denotes the divergence of the endomorphism field SA_{F}; see Section 2 for details.

In Section 3 we consider hypersurfaces with bounded F -mean curvature spanning^{1} a given Jordan curve Γ ⊂ ℝ^{n+1}, i.e., we take an immersion X : M → ℝ^{n+1} mapping the boundary ∂M topologically onto Γ.

A parametric Lagrangian F (y, z) is said to be (uniformly) elliptic, if there exists a constant M_{1} > 0 such that

|z|⟨ζ, F_{zz}(y, z)ζ⟩ ≥ M_{1}|ζ^{tan}|^{2} (E) for all (y, z) ∈ ℝ^{n+1} × (ℝ^{n+1}\{0}), ζ ∈ ℝ^{n+1}, where ζ^{tan} := ζ − ⟨ζ, z⟩z/|z|^{2}. Notice that A(z) and E(y, z) as defined in (1.4) and (1.5) are elliptic satisfying (E) with M_{1} = 1.

1The existence of conformally parametrized F-minimizing surfaces under Plateau type boundary conditions was proven in [14] and [15] for n = 2 and arbitrary co-dimension, but these solutions might have branch points. For the restricted class of boundary contours considered in Theorem 1.2, White [24] has constructed an embedded F-minimizing disk in ℝ^{3}.

Surfaces of vanishing F -mean curvature, where F is elliptic, have the convex hull property as proven in [2, Thm. 2.3]:

Theorem 1.2. Let F = F (z) ∈ C^{0}(ℝ^{n+1}) ∩ C^{3}(ℝ^{n+1}\{0}) be a parametric Lagrangian satisfying (E). Suppose X ∈ C^{0}(\overline{M}, ℝ^{n+1}) ∩ C^{2}(M, ℝ^{n+1}) is an immersion of vanishing F -mean curvature, i.e., with H_{F} (X, N) = H_{F} (X) ≡ 0. If X spans a Jordan curve Γ ⊂ ℝ^{n+1} contained in the boundary of a closed convex set K ⊂ ℝ^{n+1}, then X(\overline{M}) ⊂ K.

For surfaces of bounded (but not necessarily vanishing) F -mean curvature spanning a given Jordan curve Γ within the infinite cylinder

Z_{h} := \{(x^{1}, …, x^{n+1}) ∈ ℝ^{n+1} : h\sqrt{(x^{1})^{2} + ⋯ + (x^{n})^{2}} ≤ 1\}, h ≥ 0, (1.16)

we restrict our attention to Jordan curves Γ ⊂ ℝ^{n+1} with an orthogonal projection onto an h-convex domain \overline{Ω} ⊂ B_{h^{−1}}(0) ⊂ ℝ^{n}. Following Sauvigny [21] we call a bounded convex domain Ω ⊂ ℝ^{n} κ-convex for some κ > 0, if for every w_{0} ∈ ∂Ω there is a point ξ_{0} = ξ_{0}(w_{0}) ∈ ℝ^{n} such that the ball B_{1/κ}(ξ_{0}) ⊂ ℝ^{n} contains Ω and such that w_{0} ∈ ∂B_{1/κ}(ξ_{0}).

Theorem 1.3. Let F = F (y, z) ∈ C^{0}(ℝ^{n+1}×ℝ^{n+1})∩C^{3}(ℝ^{n+1}×(ℝ^{n+1}\{0})) be a parametric Lagrangian satisfying (E). Suppose X : \overline{M} → Z_{h} of class C^{0}(\overline{M}, ℝ^{n+1}) ∩ C^{2}(M, ℝ^{n+1}) is an immersion of prescribed F -mean curvature H_{F} ∈ C^{0}(ℝ^{n+1}), where H_{F} (y) satisfies

∥H_{F} ∥_{C^{0}(ℝ^{n+1})} ≤ M_{1}h(n − 1), (1.17)

and X spans a curve Γ ⊂ Z_{h}, whose orthogonal projection onto ℝ^{n} lies in an h-convex domain Ω ⊂ B_{h^{−1}}(0) ⊂ ℝ^{n}. Then

X(B) ⊂ Z_{Ω} := {(x^{1}, …, x^{n+1}) ∈ ℝ^{n+1} : (x^{1}, …, x^{n}) ∈ Ω}. (1.18)

In general one cannot expect that surfaces of bounded F -mean curvature satisfying the conditions of Theorem 1.3 can be represented as a graph over the h-convex domain Ω ⊂ ℝ^{n}. For n = 2 and stable surfaces of bounded mean curvature H(y) ∈ C^{1,α}(ℝ^{3}), however, Sauvigny was able to prove such a result [21] under a sign condition on \frac{∂}{∂y^{3}} H, and it turns out that the same is true for stable surfaces of prescribed F -mean curvature in ℝ^{3}; see Theorem 1.4 below.

Before defining stability in Section 4 we generalize do Carmo’s [1] second variation formula for the area functional (1.6) to the parametric functional (1.12). That is, we derive a general formula for the second variation δ^{2}F^{0}(X, Ξ) of the functional (1.12) at critical immersions X : M → ℝ^{n+1} in the direction of an arbitrary compactly supported vector field Ξ ∈ C_{0}^{2}(M, ℝ^{n+1}) containing normal and tangential terms^{2} ; see Theorem 4.1. For immersions X : M → ℝ^{n+1} of prescribed F -mean curvature H_{F}, however, the tangential term drops out (see Corollary 4.2), which additionally implies a simplified differential equation for the normal N of such surfaces derived in Corollary 4.3:

Δ_{F} N + [ tr (A_{F} S^{2}) − ⟨∇_{ℝ^{n+1}}H_{F} (X), N⟩]N = −∇_{ℝ^{n+1}}H_{F} (X). (1.19)

By means of this equation we are able to generalize Sauvigny’s result [21] for surfaces of bounded mean curvature mentioned above to stable surfaces of prescribed F -mean curvature in ℝ^{3}:

Theorem 1.4. Let F = F (z) ∈ C^{0}(ℝ^{3})∩C^{3}(ℝ^{3}\{0}) be an elliptic parametric Lagrangian satisfying (E). Suppose X : \overline{B} → Z_{h} is of class C^{3}(B, ℝ^{3}) ∩ C^{1,α}(\overline{B}, ℝ^{3}) for some α ∈ (0, 1) and a stable immersion of prescribed F - mean curvature H_{F} ∈ C^{1,α}(ℝ^{3}), where H_{F} satisfies

∥H_{F} ∥_{C^{0}(ℝ^{3})} ≤ M_{1}h. (1.20)

We assume that X spans Γ ⊂ Z_{h}, where Γ is a Jordan curve given as a graph over the boundary ∂Ω of an h-convex domain Ω ⊂ ℝ^{2}. Then X(B) ⊂ Z_{Ω}, and X(\overline{B}) can be represented as a graph over Ω, if \frac{∂}{∂y^{3}} H_{F} (y) ≥ 0 for all y = (y^{1}, y^{2}, y^{3}) ∈ ℝ^{3}.

The proof of this result can be found in Section 5. For minimal surfaces this result is due to Radó [19]. Gulliver and Spruck [13] generalized Radó’s theorem to surfaces of constant mean curvature.

Remark. For simplicity of presentation we have assumed throughout this paper that the surfaces are immersed up to the boundary. The strong smoothness hypotheses of Theorem 1.4, however, allow us to exclude boundary branch points for the specific boundary configurations considered in Theorems 1.2 and 1.4 with n = 2; see the corresponding remarks in Sections 3 and 5. That is, a conformally parametrized surface of class C^{1,α}(\overline{B}, ℝ^{3}) without interior branch points does not have boundary branch points if it either has vanishing F -mean curvature with boundary contour Γ ⊂ ∂K for some convex set K ⊂ ℝ^{3}, or if it has prescribed F -mean curvature H_{F} satisfying (1.20), with boundary contour Γ ⊂ Z_{h} as in Theorem 1.4.

A general boundary regularity result, however, guaranteeing C^{1,α}-smooth-ness up to the boundary is currently only available for F-minimizers; see [16], but not for F-critical points.

2Theorem 4.1 contains as special cases the corresponding second variation formulas of Sauvigny [21] and Räwer [20] who consider only normal, or F -normal variations, respectively.

2. Preliminaries and a differential equation for the normal

In terms of the induced metric g : T_{w}M × T_{w}M → ℝ defined in (1.14) we can express an arbitrary tangent vector V ∈ T_{w}M as

V = g^{kj}g(V, ∂_{j})∂_{k}, (2.1)

and its image under the isomorphism DX : T_{w}M → T_{X(w)}M, where M := X(\overline{M}) ⊂ ℝ^{n+1}, as

DX(V) = g^{kj}g(V, ∂_{j})∂_{k}X. (2.2)

Here g^{kj} are the coefficients of the inverse of the metric tensor g_{kj} and

\{∂_{1}, …, ∂_{n}\} := \{\frac{∂}{∂x^{1}} , …, \frac{∂}{∂x^{n}} \}

is the coordinate basis spanning T_{w}M. Let χ(M) be the space of vector fields of class C^{2} on M and denote by ∇_{V} the covariant derivative in the direction of V ∈ χ(M). We set ∇_{i} := ∇_{∂_{i}}, i = 1, …, n. We will frequently use the following versions of the product rule

U(g(V, W)) = g(∇_{U}V, W) + g(V, ∇_{U}W), (2.3) ∇_{U}(AV) = (∇_{U}A)V + A(∇_{U}V), (2.4)

for all U, V, W ∈ χ(M) and all differentiable endomorphism fields A ∈ End (T M). As a consequence of (2.3) we obtain for symmetric A ∈ End (T M) and ϕ ∈ C^{2}(M)

U(dϕ(AV)) = g(∇_{U}(A grad ϕ), V) + dϕ(A∇_{U}V), (2.5)

where g( grad ϕ, V) := dϕ(V), V ∈ T_{w}M, defines the gradient of the function ϕ on M as usual. Using the fact that ⟨DX(V), N⟩ = 0 one can show that

U(DX(V)) = DX(∇_{U}V) − ⟨DX(V), DX ∘ S(U)⟩N (2.6)

for all U, V ∈ χ(M). The trace of an endomorphism A ∈ End (T M) in local coordinates is given by

tr A = g^{ik}g(A∂_{i}, ∂_{k}). (2.7)

In particular, we will denote

(2.8) tr (A∇_{•}V) = g^{ik}g(A∇_{i}V, ∂_{k}) for A ∈ End (T M), V ∈ χ(M).

For A := Id we obtain the usual divergence of a vector field W ∈ χ(M)

(2.9) div W = div_{M}W := tr (∇_{•}W) = g^{ik}g(∇_{i}W, ∂_{k}).

The divergence Div Z of a (not necessarily tangential) vector field Z : M → ℝ^{n+1} is given in local coordinates by

Div Z = g^{ik}⟨DZ(∂_{i}), DX(∂_{k})⟩. (2.10)

For Z := DX(W), W ∈ χ(M), we get Div Z = div W by (2.9) and (2.10). We will also use the notion of the divergence of an endomorphism field div A, A ∈ End (T M), with adjoint A^{∗}, given by

g( div A, V) := tr (∇_{•}A^{∗}V) = g^{ik}g((∇_{i}A^{∗})V, ∂_{k}). (2.11)

In local coordinates we can write

div A = g^{ik}(∇_{i}A)∂_{k}, (2.12)

where ∇_{i}A denotes the covariant derivative of the tensor A; see [9, Def. 2.60].

If we denote the coefficients of the second fundamental form of (M, g) with h_{ij} := −g(∂_{i}, S(∂_{j})), and, correspondingly, the coefficients of the F -second fundamental form by

h_{F ij} := −g(∂_{i}, A_{F} S(∂_{j})) = −⟨F_{zz}∂_{i}X, ∂_{j}N⟩, (2.13)

then the F -mean curvature H_{F} defined in (1.9) can be written as

H_{F} = − tr (A_{F} S) = −g^{ij}g(∂_{i}, A_{F} S(∂_{j})) = g^{ij}h_{F ij}. (2.14)

Introducing the second order differential operator

Θ_{F} := Δ_{F} − div A_{F} , (2.15)

where Δ_{F} is given by (1.13), the first author could prove in [2] that

Θ_{F} X = H_{F} N (2.16)

holds for any immersion X ∈ C^{2}(M, ℝ^{n+1}). This equation reduces to the classical identity ΔX = div grad X = HN, if F (y, z) = A(z), or F (y, z) = E(y, z), respectively; see (1.4), (1.5). Moreover, Θ_{F} is uniformly elliptic if F satisfies the ellipticity condition (E), which leads to the enclosure theorems proven in [2], and which will be used in the proofs of Sections 3 and 5.

Now we will conclude this section with the

Proof of Theorem 1.1: Apply (2.5) to ϕ := N^{i}, i = 1, …, n+1, and A := A_{F} ∈ End (T M) to obtain by (2.6) and (2.4)

g(∇_{U}(A_{F} grad N), V) =_{(2.5)} U(DN(A_{F} V)) − DN(A_{F} ∇_{U}V)
= U(DX(SA_{F} V)) − DX(SA_{F} ∇_{U}V)
=_{(2.6)} DX(∇_{U}(SA_{F} V)) − ⟨DX(SA_{F} V), DX ∘ S(U)⟩N
−DX(SA_{F} ∇_{U}V)
=_{(2.4)} DX(∇_{U}(SA_{F})V) − g(SA_{F} (V), S(U))N.

Choosing U = ∂_{i}, V = ∂_{k} we obtain by (1.13), (2.9), (2.12) and (2.7)

Δ_{F} N =_{(1.13)} div (A_{F} grad N)
=_{(2.9)}  g^{ik}g(∇_{i}(A_{F} grad N), ∂_{k})
=  g^{ik}DX(∇_{i}(SA_{F})∂_{k}) − g^{ik}g(SA_{F} (∂_{k}), S(∂_{i}))N
=_{(2.12),(2.7)} DX( div (SA_{F})) − tr (A_{F} S^{2})N,

where we have used the symmetry of A_{F} and S to obtain the last term. □ Using the Codazzi equation (cf. [18, p. 30])

(∇_{V} S)W = (∇_{W} S)V (2.17)

one can show that

div S = − grad H. (2.18)

Thus in case of the functionals (1.6) or (1.7), where A_{F} is the identity, we obtain

ΔN + tr (S^{2})N = −DX( grad H) .

Therefore (1.15) is a generalization of [21, Hilfssatz 1].

3. Proofs of the enclosure theorems

For the convenience of the reader we recall the proof of Theorem 1.2 from [2].

Proof of Theorem 1.2: Since H_{F} (X, N) = H_{F} (X) = 0, we infer from (2.16) that

Θ_{F} (t(X)) = 0 (3.1)

for all affine linear functions

t(y) := ⟨a, y⟩ + b, a ∈ ℝ^{n+1}, b ∈ ℝ. (3.2)

Taking an arbitrary supporting half plane of the convex body K characterized by an affine linear function t_{K}, we have t_{K}(X) ≤ 0 on ∂M, and hence by (3.1) and the maximum principle [11, p. 32], t_{K}(X) ≤ 0 on \overline{M}, i.e., X(\overline{M}) ⊂ K. □

Remark. For n = 2, M := B = B_{1}(0) ⊂ ℝ^{2}, and X immersed only in the interior of B but given in conformal parameters, i.e. with

|X_{u}|^{2} = |X_{v}|^{2} and ⟨X_{u}, X_{v}⟩ = 0 on \overline{B}, (3.3)

we can exclude boundary branch points. In fact, introducing polar coordinates (r, ϑ) in B and fixing w_{0} ∈ ∂B we can apply Hopf’s boundary point lemma [11, p. 34] together with (3.1) to obtain for t : ℝ^{3} → ℝ as in (3.2)

\frac{∂}{∂ν} [t(X(w))]_{w=w_{0}} = ⟨a, X_{r}(w_{0})⟩ > 0.

Therefore we have |X_{r}(w_{0})| > 0. Rewriting (3.3) in polar coordinates we conclude |X_{ϑ}(w_{0})| > 0 which shows that w_{0} is not a branch point.

Proof of Theorem 1.3: For the function R(x) := (x^{1})^{2} + ⋯ + (x^{n})^{2} we compute similarly as in [5, p. 7] using (2.1), (2.15), (2.16), (2.12) and (E)

\frac{1}{2}Θ_{F} (R(X)) =_{(2.1) {(2.15) \atop (2.12)}}  ∑_{i=1}^{n} X^{i} div (A_{F} grad (X^{i}))

+ ∑_{i=1}^{n} g( grad (X^{i}), A_{F} grad (X^{i})) − ∑_{i=1}^{n} X^{i}( div A_{F})(X^{i})

=_{(2.15)} ∑_{i=1}^{n} X^{i}Θ_{F} (X^{i}) + ∑_{i=1}^{n} g( grad (X^{i}), A_{F} grad (X^{i}))

≥_{{(2.16) \atop (E)}}  ∑_{i=1}^{n} H_{F} (X, N)X^{i}N^{i} + M_{1} ∑_{i=1}^{n} g( grad (X^{i}), grad (X^{i}))

≥ −|H_{F} (X)|\sqrt{R(X)} + M_{1}(n − 1)
≥  M_{1}(n − 1) − ∥H_{F} ∥_{C^{0}(ℝ^{3})}h^{−1}

on M, since X(\overline{M}) ⊂ Z_{h}; see (1.16). Notice that we have used the relation p^{ii} = g( grad (X^{i}), grad (X^{i})), where P = P (w) = (p^{ij})(w) : ℝ^{n+1} → T_{X(w)}M is the orthogonal projection onto the n-dimensional tangent plane of M = X(\overline{M}), with ∑_{i=1}^{n+1} p^{ii} = n, so that

∑_{i=1}^{n} g( grad (X^{i}), grad (X^{i})) = n − g( grad (X^{n+1}), grad (X^{n+1})) ≥ n − 1.

Thus Θ_{F} (R(X)) ≥ 0 on M due to (1.17), and the maximum principle implies R(X(w)) < h^{−2} for all w ∈ M, since R(X) ≢ h^{−2} in M. Following Sauvigny [21] we now argue as follows. Assuming that there is some point w^{∗} ∈ M with X(w^{∗}) ∉ Z_{Ω} we infer that x^{∗} := (X^{1}(w^{∗}), X^{2}(w^{∗})) ∉ Ω. Let y^{∗} ∈ ∂Ω be a point with |y^{∗} − x^{∗}| = dist (x^{∗}, Ω) ≥ 0. (If x^{∗} ∈ ∂Ω take y^{∗} := x^{∗}.) Since Ω is h-convex there is a point η_{∗} ∈ ℝ^{n} such that Ω ⊂ B_{1/h}(η_{∗}) and y^{∗} ∈ ∂B_{1/h}(η_{∗}). Thus X(w^{∗}) ∉ Z_{B_{1/h}(η_{∗})} and we can look at the 1-parameter family of cylinders {Z(λ) := Z_{B_{1/h}(λη_{∗})}}_{0≤λ≤1}, for which

X(\overline{B}) ⊂ Z(0) = Z_{h},

and

X(B) ∩ ∂Z(1) = X(B) ∩ ∂Z_{B_{1/h}(η_{∗})} ≠ ∅.

By continuity we find λ_{0} ∈ [0, 1] with

X(\overline{B}) ⊂ Z(λ_{0}) and X(B) ∩ ∂Z(λ_{0}) ≠ ∅. (3.4)

With the same computation as before we deduce for R_{0}(x) := (x^{1} −λ_{0}η_{∗}^{1})^{2} + ⋯ + (x^{n} − λ_{0}η_{∗}^{n})^{2} the inequality Θ_{F} (R_{0}(X)) ≥ 0 on B; hence by (3.4) and the maximum principle R_{0}(X(w)) ≡ h^{−2}, which is absurd. Thus we have shown (1.18). □

4. A general second variation formula and stability

In this section we consider C^{3}-perturbations X(., ϵ) : M × (−ϵ_{0}, ϵ_{0}) → ℝ^{n+1} of an immersed hypersurface X ∈ C^{3}(M, ℝ^{n+1}) with

X(., 0) = X and (4.1)

\frac{∂}{∂ϵ}X(., ϵ)_{ϵ=0} = φN + DX(V) =: Ξ, (4.2)

where φ ∈ C_{0}^{2}(M), V ∈ χ(M) with compact support. Notice that we admit a non-vanishing tangential component in the variational field Ξ ∈ C_{0}^{2}(M, ℝ^{n+1}) as in [1] but in contrast to [21, p. 64]. The second variation δ^{2}F^{0}(X, Ξ) of the functional F^{0} defined in (1.12) at X in the direction of Ξ is defined as

δ^{2}F^{0}(X, Ξ) := \frac{d^{2}}{dϵ^{2}} F^{0}(X(., ϵ))_{ϵ=0}. (4.3)

Theorem 4.1. Let F = F (z) ∈ C^{0}(ℝ^{n+1}) ∩ C^{3}(ℝ^{n+1}\{0}) be a parametric Lagrangian. Suppose X ∈ C^{3}(M, ℝ^{n+1}) is a critical immersion for the functional (1.12) and Ξ ∈ C_{0}^{2}(M, ℝ^{n+1}) is a variational field of the form (4.2). Then

δ^{2}F^{0}(X, Ξ) = ∫_{M} {g(A_{F} grad φ, grad φ) − φ^{2}( tr (A_{F} S^{2})

−⟨∇_{ℝ^{n+1}}H_{F} (X), N⟩) +φg( div (SA_{F}) + grad H_{F} (X), V)} dA. (4.4)

Note that only first order derivatives of X(., ϵ) with respect to ϵ, i.e., merely Ξ defined in (4.2) enters the formula for the second variation which justifies the notation on the left-hand side of (4.3).

Proof of Theorem 4.1: Using the identity

\frac{∂}{∂η} X(., ϵ + η)_{η=0} = \frac{∂}{∂ϵ̄}X(., ϵ̄)_{ϵ̄=ϵ}

we obtain

δ^{2}F^{0}(X, Ξ) = \frac{d^{2}}{dϵ^{2}} F^{0}(X(., ϵ))_{ϵ=0}
= \frac{d}{dϵ} (\frac{d}{dη} F^{0}(X(., ϵ + η))_{η=0}) _{ϵ=0}
= \frac{d}{dϵ} (δF^{0}(X(., ϵ), \frac{∂}{∂ϵ̄}X(., ϵ̄)_{ϵ̄=ϵ})) _{ϵ=0}.

Hence, by the first variation formula proved in [2, pp. 5,6] applied to (1.12) and evaluated at X(., ϵ) in the direction \frac{∂}{∂ϵ̄}X(., ϵ̄)_{ϵ̄=ϵ},

δ^{2}F^{0}(X, Ξ) = \frac{d}{dϵ} (∫_{M} \{ ⟨\frac{∂}{∂ϵ̄}X(., ϵ̄)_{ϵ̄=ϵ}, N(., ϵ)⟩

[H_{F} (X(., ϵ)) − H_{F} (X(., ϵ), N(., ϵ))]} dA) _{ϵ=0}, (4.5)

where N(., ϵ) is the unit normal and H_{F} (X(., ϵ), N(., ϵ)) the F -mean curvature of the perturbed immersion X(., ϵ) ∈ C^{3}(M, ℝ^{n+1}).

According to [2, Lemma 1.1] one has

\frac{∂}{∂ϵ}N(., ϵ)_{ϵ=0} = −DX( grad φ) + DN(V), (4.6)

where φ ∈ C_{0}^{2}(M) and V ∈ χ(M) with compact support determine the normal and tangential component of Ξ defined in (4.2). From (2.14), on the other hand, we infer

\frac{∂}{∂ϵ}H_{F} (X(., ϵ), N(., ϵ))_{ϵ=0} =_{(2.14)} [(\frac{∂}{∂ϵ}g^{ij}(ϵ)) h_{F ij} + g^{ij} (\frac{∂}{∂ϵ}h_{F ij}(ϵ))] _{ϵ=0}
(4.7)  =: I + II,

where the argument ϵ indicates that the corresponding quantity belongs to the perturbed immersion X(., ϵ). In particular, we write, e.g., g^{ij}(0) = g^{ij}, ∂_{l}X(., 0) = ∂_{l}X, etc. On account of g^{ij}(ϵ)g_{js}(ϵ) = δ_{s}^{i} for all ϵ ∈ (−ϵ_{0}, ϵ_{0}) one has

\frac{∂}{∂ϵ}g^{ij}(ϵ) = −g^{ik}(ϵ) (\frac{∂}{∂ϵ}g_{kl}(ϵ)) g^{lj}(ϵ),

and therefore by (4.2) and (2.6) for U := ∂_{k}, ∂_{l}, respectively,

\frac{∂}{∂ϵ}g^{ij}(ϵ)_{ϵ=0} = −g^{ik} \{⟨\frac{∂}{∂ϵ}(∂_{k}X(., ϵ)), ∂_{l}X⟩ + ⟨∂_{k}X, \frac{∂}{∂ϵ}(∂_{l}X(., ϵ))⟩\} _{ϵ=0}g^{lj}

=_{(4.2)} −g^{ik} {⟨φ∂_{k}N + ∂_{k}(DX(V)), ∂_{l}X⟩ + ⟨∂_{k}X, φ∂_{l}N + ∂_{l}(DX(V))⟩} g^{lj}

=_{(2.6)} 2φg^{ik}h_{kl}g^{lj} − g^{ik}⟨DX(∇_{k}V), ∂_{l}X⟩g^{lj} − g^{ik}⟨DX(∇_{l}V), ∂_{k}X⟩g^{lj}.

12 U. CLARENZ AND H. VON DER MOSEL

Thus we obtain for the expression I in (4.7) by the symmetry of the mappings A_{F} and S

I = (\frac{∂}{∂ϵ}g^{ij}(ϵ)) _{ϵ=0}h_{F ij}
= 2φg^{ik}h_{kl}g^{lj}h_{F ij} − g^{ik}⟨DX(∇_{k}V), ∂_{l}X⟩g^{lj}h_{F ij}
−g^{ik}⟨DX(∇_{l}V), ∂_{k}X⟩g^{lj}h_{F ij}
=_{(2.13)} 2φg^{ik}g(∂_{k}, S(∂_{l}))g^{lj}g(∂_{i}, A_{F} S(∂_{j}))
+g^{ik}g(∇_{k}V, ∂_{l})g^{lj}g(∂_{i}, A_{F} S(∂_{j}))
+g^{ik}g(∇_{l}V, ∂_{k})g^{lj}g(∂_{i}, A_{F} S(∂_{j}))
= 2φg^{ik}g(S(∂_{k}), g^{lj}g(SA_{F} (∂_{i}), ∂_{j})∂_{l})
+g^{ik}g(SA_{F} (∂_{i}), g^{lj}g(∇_{k}V, ∂_{l})∂_{j})
+g^{lj}g(g^{ik}g(∇_{l}V, ∂_{k})∂_{i}, A_{F} S(∂_{j}))
=_{(2.1)} 2φg^{ik}g(A_{F} S^{2}(∂_{k}), ∂_{i}) + g^{ik}g(SA_{F} (∂_{i}), ∇_{k}V)
+g^{lj}g(∇_{l}V, A_{F} S(∂_{j}))
(4.8)  =_{{(2.7) \atop (2.8)}} 2φ tr (A_{F} S^{2}) + tr ((SA_{F} + A_{F} S)∇_{•}V).

Furthermore we need to compute

\frac{∂}{∂ϵ}h_{F ij}(ϵ)_{ϵ=0} =_{(2.13)} − ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ))_{ϵ=0}∂_{i}X, ∂_{j}N⟩

− ⟨F_{zz}(N)∂_{i} [\frac{∂}{∂ϵ}X(., ϵ)_{ϵ=0}] , ∂_{j}N⟩

− ⟨F_{zz}(N)∂_{i}X, ∂_{j} [\frac{∂}{∂ϵ}N(., ϵ)_{ϵ=0}]⟩

=_{(4.2),(4.6)} − ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ))_{ϵ=0}∂_{i}X, ∂_{j}N⟩

− ⟨F_{zz}(N)∂_{i} [φN + DX(V)] , ∂_{j}N⟩ (4.9)

− ⟨F_{zz}(N)∂_{i}X, ∂_{j} [−DX( grad φ) + DN(V)]⟩ .

Since ⟨∂_{j}N, N⟩ = 0 we have by (2.6)

⟨F_{zz}(N)∂_{i}[φN + DX(V)], ∂_{j}N⟩ = φ⟨F_{zz}(N)∂_{i}N, ∂_{j}N⟩

+⟨F_{zz}(N)DX(∇_{i}V), ∂_{j}N⟩, (4.10)

ON SURFACES OF PRESCRIBED F -MEAN CURVATURE 13

and also by (2.6) and on account of DN = DX ∘ S ⟨F_{zz}(N)∂_{i}X, ∂_{j}[−DX( grad φ) + DN(V)]⟩ (4.11)

= −⟨F_{zz}(N)∂_{i}X, ∂_{j}DX( grad φ)⟩ + ⟨F_{zz}(N)∂_{i}X, ∂_{j}DX(S(V))⟩
=_{(2.6)} −⟨F_{zz}(N)∂_{i}X, ∂_{j}DX( grad φ)⟩ + ⟨F_{zz}(N)∂_{i}X, DX(∇_{j}(S(V))⟩.

Inserting (4.9)–(4.11) into the expression for II in (4.7) leads to

II = −g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ)))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ − φg^{ij}⟨F_{zz}(N)∂_{i}N, ∂_{j}N⟩

−g^{ij}⟨F_{zz}(N)DX(∇_{i}V), ∂_{j}N⟩ + g^{ij}⟨F_{zz}(N)∂_{i}X, ∂_{j}DX( grad φ)⟩

−g^{ij}⟨F_{zz}(N)∂_{i}X, DX(∇_{j}(S(V))⟩
=_{(1.10)} −g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ)))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ − φg^{ij}g(A_{F} S(∂_{i}), S(∂_{j}))
−g^{ij}g(A_{F} (∇_{i}V), S(∂_{j})) + g^{ij}⟨F_{zz}(N)∂_{i}X, ∂_{j}DX( grad φ)⟩
−g^{ij}g(A_{F} (∂_{i}), ∇_{j}(S(V))).

By the symmetry of A_{F} and S and by (2.7) (and (2.4) for the last term) we may rewrite this as

II = −g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ)))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ − φ tr (A_{F} S^{2})

− tr (SA_{F} ∇_{•}V) + g^{ij}⟨F_{zz}(N)∂_{i}X, ∂_{j}DX( grad φ)⟩ (4.12)

− tr (A_{F} S∇_{•}V) − tr (A_{F} ∘ [(∇_{•}S)V]).

Adding (4.8) and (4.12) in (4.7) and using the symmetry of F_{zz} we arrive at

\frac{∂}{∂ϵ}H_{F} (X(., ϵ), N(., ϵ))_{ϵ=0}

=_{(4.8),(4.12)} φ tr (A_{F} S^{2}) + g^{ij}⟨∂_{i}X, F_{zz}(N)∂_{j}DX( grad φ)⟩

−g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ)))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ − tr (A_{F} ∘ [(∇_{•}S)V]). (4.13)

By virtue of (1.10), (2.6), (2.9) and (1.13) we may rewrite the second term on the right-hand side as

g^{ij}⟨∂_{i}X, F_{zz}(N)∂_{j}DX( grad φ)⟩

= g^{ij}⟨∂_{i}X, ∂_{j}{F_{zz}(N)DX( grad φ)}⟩

−g^{ij}⟨∂_{i}X, ∂_{j}(F_{zz}(N))DX( grad φ)⟩ (4.14)

=_{{(1.10) \atop (2.6)}} g^{ij}g(∂_{i}, ∇_{j}(A_{F} ( grad φ))) − g^{ij}⟨∂_{i}X, ∂_{j}(F_{zz}(N))DX( grad φ)⟩
=_{{(2.9) \atop (1.13)}} Δ_{F} φ − g^{ij}⟨∂_{i}X, ∂_{j}(F_{zz}(N))DX( grad φ)⟩.

14 U. CLARENZ AND H. VON DER MOSEL

Moreover, by the symmetry of F_{zz} we have

g^{ij}⟨∂_{i}X, ∂_{j}(F_{zz}(N))DX( grad φ)⟩ = g^{ij}⟨∂_{j}(F_{zz}(N))∂_{i}X, DX( grad φ)⟩, (4.15)

and by (2.6), (1.10), (2.4) and (2.11) for general W ∈ T_{w}M

g^{ij}⟨∂_{i}X, ∂_{j}(F_{zz}(N))DX(W)⟩

=_{(2.6)} g^{ij}⟨∂_{j}(F_{zz}(N)DX(∂_{i})) − F_{zz}(N)DX(∇_{j}∂_{i}), DX(W)⟩
=_{(1.10)} g^{ij}⟨∂_{j}(DX(A_{F} (∂_{i}))) − DX(A_{F} (∇_{j}∂_{i})), DX(W)⟩
=_{{(2.4) \atop (2.6)}} g^{ij}⟨DX((∇_{j}A_{F})∂_{i}), DX(W)⟩
(4.16) = g^{ij}g((∇_{j}A_{F})∂_{i}, W) =_{(2.12)} g( div A_{F} , W).

Summarizing (4.13), (4.14) and (4.16) for W := grad φ we arrive at

\frac{∂}{∂ϵ}H_{F} (X(., ϵ), N(., ϵ))_{ϵ=0} = Δ_{F} φ + φ tr (A_{F} S^{2}) − g( div A_{F} , grad φ)

−g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ (4.17)

− tr (A_{F} ∘ [(∇_{•}S)V]).

Now writing out components one calculates

\frac{∂}{∂ϵ}(F_{z^{l}z^{k}}(N(., ϵ)))_{ϵ=0}∂_{i}X^{k}∂_{j}N^{l} = F_{z^{l}z^{k}z^{s}}(N) (\frac{∂}{∂ϵ}N^{s}(., ϵ)_{ϵ=0}) ∂_{i}X^{k}∂_{j}N^{l}
= ∂_{j}(F_{z^{k}z^{s}}(N)) (\frac{∂}{∂ϵ}N^{s}(., ϵ)_{ϵ=0}) ∂_{i}X^{k},

whence by (4.6), (4.15) and (4.16) for W := S(V) − grad φ,

g^{ij} ⟨\frac{∂}{∂ϵ}(F_{zz}(N(., ϵ)))_{ϵ=0}∂_{i}X, ∂_{j}N⟩ = g^{ij}⟨∂_{j}(F_{zz}(N))∂_{i}X, DX ∘ S(V)
−DX( grad φ)⟩
(4.18)  = g( div A_{F} , S(V) − grad φ).

Next we claim that for any V ∈ T_{w}M (4.19) g( div (SA_{F}), V) = tr (A_{F} ∘ [(∇_{•}S)V]) + g( div A_{F} , S(V)). This together with (4.18) and (4.17) leads to

\frac{∂}{∂ϵ}H_{F} (X(., ϵ), N(., ϵ))_{ϵ=0} = Δ_{F} φ + φ tr (A_{F} S^{2}) − g( div (SA_{F}), V). (4.20)

By (4.5) we then conclude using (4.2)

δ^{2}F^{0}(X, Ξ) = − ∫_{M} φ{Δ_{F} φ + φ tr (A_{F} S^{2}) − φ⟨∇_{ℝ^{n+1}}H_{F} (X), N⟩

−g( div (SA_{F}) + grad H_{F} (X), V)} dA, (4.21)

which proves Theorem 4.1. Notice that the other terms obtained by carrying out the differentiation with respect to ϵ in (4.5) and evaluating at ϵ = 0 vanish, since

H_{F} (X(., 0), N(., 0)) = H_{F} (X, N) ≡ H_{F} (X)

because X is a critical immersion for (1.12).

It remains to show (4.19). In fact, by (2.11) and the symmetry of A_{F} and S

g( div (SA_{F}), V) =_{(2.11)} g^{ik}g(∇_{i}(SA_{F})^{∗}V, ∂_{k})
= g^{ik}g((∇_{i}A_{F}^{∗})S^{∗}V, ∂_{k}) + g^{ik}g(A_{F}^{∗} (∇_{i}S^{∗})V, ∂_{k})
=_{(2.11)} g( div A_{F} , S^{∗}V) + g^{ik}g((∇_{i}S)^{∗}V, A_{F} (∂_{k}))
= g(S div A_{F} , V) + g^{ik}g(V, (∇_{i}S)A_{F} (∂_{k})).

The Codazzi equation (2.17) and the symmetry of S and A_{F} imply now

g^{ik}g(V, (∇_{i}S)A_{F} (∂_{k})) = g^{ik}g(A_{F} ∘ (∇_{i}S)V, ∂_{k}) =_{(2.8)} tr (A_{F} ∘ [(∇_{•}S)V]),

which proves the claim. □

As a consequence of Theorem 4.1 we can state

Corollary 4.2. Let X ∈ C^{3}(M, ℝ^{n+1}) be a immersion of prescribed F - mean curvature H_{F} ∈ C^{1}(ℝ^{n+1}), where F = F (z) ∈ C^{0}(ℝ^{n+1})∩C^{3}(ℝ^{n+1}\{0}) is a parametric Lagrangian. Then

(4.22) div (SA_{F}) = − grad H_{F} (X),  and
δ^{2}F^{0}(X, Ξ) = ∫_{M}{g(A_{F} grad φ, grad φ)

−[ tr (A_{F} S^{2}) − ⟨∇_{ℝ^{n+1}}H_{F} (X), N⟩]φ^{2}} dA, (4.23)

where Ξ = φN +DX(V), φ ∈ C_{0}^{2}(M), and V ∈ χ(M) with compact support. In particular, the second variation of a parametric integrand depends on normal variations only.

Proof. The symmetry argument we use here is due to White [23]. Consider the surfaces

X( · , ϵ, η) = X + ϵ(φN + DX(V)) + η(ψN + DX(W)) ,

where φ, ψ ∈ C_{0}^{∞}(M) and V, W ∈ χ(M) with compact support. Similarly as in (4.5) we have

\frac{d}{dϵ} _{ϵ=0} [\frac{d}{dη} _{η=0}F^{0}(X(., ϵ, η)]
= \frac{d}{dϵ} (∫_{M} \{ ⟨\frac{∂}{∂η} X(., ϵ, η)_{η=0}, N(., ϵ, 0)⟩

[H_{F} (X(., ϵ, 0)) − H_{F} (X(., ϵ, 0), N(., ϵ, 0))]} dA) _{ϵ=0}. (4.24)

Hence by (4.20) we obtain

\frac{d}{dϵ} _{ϵ=0} [\frac{d}{dη} _{η=0}F^{0}(X(., ϵ, η)]
= ∫_{M} ψ{−Δ_{F} φ − φ tr (A_{F} S^{2}) + g( div (SA_{F}), V)}
+ψ⟨∇_{ℝ^{n+1}}H_{F} (X), φN + DX(V)⟩ dA.

Since

\frac{d}{dϵ} _{ϵ=0} [\frac{d}{dη} _{η=0}F^{0}(X(., ϵ, η)] = \frac{d}{dη} _{η=0} [\frac{d}{dϵ} _{ϵ=0}F^{0}(X(., ϵ, η)] ,

we arrive at

∫_{M} ψg( div (SA_{F}), V) + ψg(V, grad H_{F} (X)) dA
(4.25)  = ∫_{M} φg( div (SA_{F}), W) + φg(W, grad H_{F} (X)) dA

for all φ, ψ ∈ C_{0}^{∞}(M) and V, W ∈ χ(M) with compact support, where we used that

∫_{M} ψ(−Δ_{F} φ − φ tr (A_{F} S^{2})) dA = ∫_{M} φ(−Δ_{F} ψ − ψ tr (A_{F} S^{2})) dA.

Equation (4.25) is only possible if div (SA_{F}) = − grad H_{F} (X), for if not, we could choose W ≡ 0 to have a vanishing right-hand side in (4.25), and ψ and V appropriately to obtain a positive left-hand side and thus a contradiction.

□

Inserting (4.22) into formula (1.15) of Theorem 1.1 for the normal of an F^{0}-critical immersion we obtain

Corollary 4.3. Let N be the normal of an immersion X ∈ C^{3}(M, ℝ^{n+1}) of prescribed F -mean curvature H_{F} (y) ∈ C^{1}(ℝ^{n+1}), where F = F (z) is a parametric Lagrangian of class C^{0}(ℝ^{n+1}) ∩ C^{3}(ℝ^{n+1}\{0}). Then

Δ_{F} N + [ tr (A_{F} S^{2}) − ⟨∇_{ℝ^{n+1}}H_{F} (X), N⟩]N = −∇_{ℝ^{n+1}}H_{F} (X). (4.26)

The above corollary generalizes [21, Satz 1].

The notion of stability is defined as follows.

Definition 4.4. Let X ∈ C^{3}(M, ℝ^{n+1}) be an F^{0}-critical immersion, where F^{0} is defined in (1.12) with a parametric Lagrangian F = F (z) ∈ C^{0}(ℝ^{n+1})∩ C^{3}(ℝ^{n+1}\{0}). Then X is called stable if δ^{2}F^{0}(X, Ξ) ≥ 0 for all Ξ ∈ C_{0}^{2}(M, ℝ^{n+1}). If δ^{2}F^{0}(X, Ξ) > 0 we say X is strictly stable.

5. Graph representation of prescribed F -mean curvature surfaces

The proof of Theorem 1.4 is based on a maximum principle for elliptic equations of the form Lu = (α^{ij}u_{x^{j}})_{x^{i}} + β^{i}u_{x^{i}} + cu. Usually it is required that the coefficient c be non-positive. As was carried out in [21] for the Laplace operator this condition may be replaced by assuming that the first eigenvalue of L is non-negative. Our proof of the corresponding lemma for general elliptic equations is related to [12, Lemma 1], but we assume less regularity of the coefficients:

Lemma 5.1. Let Lu = (α^{ij}u_{x^{j}})_{x^{i}}+β^{i}u_{x^{i}}+cu ≤ 0 be a linear elliptic equation in a domain Ω ⊂ ℝ^{n} with smooth boundary, where α^{ij}, β^{i}, c ∈ C^{0,μ}(\overline{Ω}) and with α^{ij} = α^{ji} for i, j = 1, …, n. Assume that the first eigenvalue of L is non-negative on Ω. If for u ∈ C^{2}(Ω) ∩ C^{0}(\overline{Ω}) we have Lu ≤ 0 and u_{|∂Ω} > 0, then inf_{Ω} u > 0.

Proof. On account of the continuity of u we can assume that there is a smoothly bounded domain Ω_{2} ⊂⊂ Ω with u_{|∂Ω_{2}} > 0. The first eigenvalue λ of L in a domain Ω_{1} with Ω_{2} ⊂⊂ Ω_{1} ⊂⊂ Ω is simple and therefore strictly positive, since otherwise we could apply [11, Thm. 8.38] to find a positive

eigenfunction ξ ∈ H^{∘}^{1,2}(Ω_{1}) on Ω_{1} with ξ = 0 on ∂Ω_{1}. Extending ξ by zero

outside of Ω_{1} we would obtain ξ̃ ∈ H^{∘}^{1,2}(Ω) with Lξ̃+ λξ̃ = 0, ξ̃ ≢ 0 on Ω with ξ̃ ≡ 0 on Ω\Ω_{1}, contradicting [11, Thm. 8.38].

The regularity of the coefficients α^{ij}, β^{i}, c, leads to ξ ∈ C^{1,μ}(Ω_{1}); see e.g. [10, Theorem 3.2]. Thus in Ω_{2} we can write u = ξ v, and due to the regularity of u and ξ we obtain a.e. on Ω_{2}:

L(ξv) = v(α^{ij}ξ_{x^{j}})_{x^{i}} + ξ(α^{ij}v_{x^{j}})_{x^{i}} + 2α^{ij}v_{x^{i}}ξ_{x^{j}} + vβ^{i}ξ_{x^{i}} + ξβ^{i}v_{x^{i}} + cvξ
= vLξ + ξ[(α^{ij}v_{x^{j}})_{x^{i}} + β^{i}v_{x^{i}}] + 2α^{ij}v_{x^{i}}ξ_{x^{j}}
= ξ[ (α^{ij}v_{x^{j}})_{x^{i}} + β^{i}v_{x^{i}} + (2/ξ) α^{ij}ξ_{x^{j}} v_{x^{i}} − λv] .

Thus we obtain an elliptic differential inequality for v:

0 ≤ ∫_{Ω_{2}}[α^{ij}v_{x^{j}}φ_{x^{i}} − (β̃^{i}v_{x^{i}} − λv)φ] dx

for all non-negative φ ∈ C_{0}^{1}(Ω_{2}), where β̃^{i} := β^{i} + (2/ξ)a^{ij}ξ_{x^{j}} ∈ L^{∞}(Ω_{2}) for i = 1, …, n. Thus the weak maximum principle [11, Thm 8.1] holds for v and we have inf_{Ω_{2}} v ≥ 0. By the strong minimum principle [11, Thm. 8.19] we obtain the strict relation inf_{Ω_{2}} v > 0. □

Proof of Theorem 1.4: According to our assumptions on Γ and Ω in Theorem 1.3 there is a function f ∈ C^{2}(∂Ω), such that Γ = {(x, f(x)) : x ∈ ∂Ω} is (positively) oriented by setting

P_{k} := (x_{k}, f(x_{k})), k = 1, 2, 3, (5.1)

where x_{k} ∈ ∂Ω, k = 1, 2, 3, are chosen in positive orientation with respect to ℝ^{2}.

Since X is immersed on \overline{B} we may assume without loss of generality that X is conformally parametrized, i.e., satisfies the conformality relations (3.3). (Otherwise we can perform a diffeomorphism w : \overline{B} → \overline{B} of class C^{2,α}(B, ℝ^{2}) ∩ C^{1,α}(\overline{B}, ℝ^{2}) such that X̃ := X ∘ w^{−1} is conformally parametrized; see e.g. [17, Corollary 3.1.2].) Performing a suitable Möbius transformation on \overline{B} we may assume that X satisfies the three-point condition

X(w_{k}) = P_{k} k = 1, 2, 3, (5.2)

where w_{1}, w_{2}, w_{3} are fixed distinct points on ∂B.

Fix some point w_{0} = e^{iϑ_{0}} ∈ ∂B. Since Ω is h-convex there is a point η_{0} ∈ ℝ^{2} such that Ω ⊂ B_{1/h}(η_{0}) and such that y_{0} := (X^{1}(w_{0}), X^{2}(w_{0})) ∈ ∂Ω is contained in ∂B_{1/h}(η_{0}). Without loss of generality we may assume that η_{0} = 0. By Hopf’s boundary point lemma we then obtain for the function R(x) := (x^{1})^{2} + (x^{2})^{2}

\frac{∂}{∂ν} R(X(w))_{w=w_{0}} > 0,

i.e., in polar coordinates (r, ϑ)

2 ∑_{i=1}^{2} X^{i}(w)\frac{∂}{∂r} X^{i}(w)_{w=w_{0}} > 0, (5.3)

which implies |X_{r}(w_{0})| > 0, and by conformality also

|X_{ϑ}(w_{0})| > 0. (5.4)

Since R(X(w_{0})) = h^{−2} ≥ R(X(w)) for all w = e^{iϑ} ∈ ∂B we obtain

0 = \frac{∂}{∂ϑ} _{ϑ=ϑ_{0}}(R(X(e^{iϑ}))) = ∑_{i=1}^{2} X^{i}(e^{iϑ})\frac{∂}{∂ϑ}X^{i}(e^{iϑ})_{ϑ=ϑ_{0}}. (5.5)

Since Γ = {(x, f(x)) : x ∈ ∂Ω}, f ∈ C^{2}(∂Ω), we have

|X_{ϑ}^{3}| = |f_{x^{1}}X_{ϑ}^{1} + f_{x^{2}}X_{ϑ}^{2}| ≤ ∥∇f̃∥_{C^{0}(ℝ^{2})}\sqrt{(X_{ϑ}^{1})^{2} + (X_{ϑ}^{2})^{2}},

where f̃ ∈ C^{2}(ℝ^{2}) is an extension of f onto ℝ^{2} with controlled C^{2}-norm; see [11, p. 137]. Hence, by (5.4),

0 < |X_{ϑ}(w)|^{2} _{w=w_{0}} ≤ (1 + ∥∇f̃∥_{C^{0}(ℝ^{2})}^{2})[(X_{ϑ}^{1}(w))^{2} + (X_{ϑ}^{2}(w))^{2}]_{w=w_{0}}. (5.6)

By (5.2) the mapping (X^{1}, X^{2}) : ∂B → ∂Ω respects the positive orientation, thus we infer from (5.5) and (5.6) that there is a constant σ > 0 such that

X_{ϑ}^{1}(w_{0}) = −σX^{2}(w_{0}), X_{ϑ}^{2}(w_{0}) = σX^{1}(w_{0}).

Therefore, by (5.3)

X_{r}^{1}(w_{0})X_{ϑ}^{2}(w_{0})−X_{r}^{2}(w_{0})X_{ϑ}^{1}(w_{0}) = σ(X^{1}(w_{0})X_{r}^{1}(w_{0})+X^{2}(w_{0})X_{r}^{2}(w_{0})) >_{(5.3)} 0,

which means that X_{u}^{1}(w_{0})X_{v}^{2}(w_{0}) − X_{v}^{1}(w_{0})X_{u}^{2}(w_{0}) > 0, i.e., the third component of N^{3} of the normal N is positive on ∂B. Moreover, by Corollary 4.3, N^{3} satisfies the elliptic differential equation

Δ_{F} N^{3} + ( tr (A_{F} S^{2}) − ⟨∇_{ℝ^{3}}H_{F} (X), N⟩)N^{3} = −\frac{∂H_{F}}{∂y^{3}} (X).

Using the assumption on H_{F} (X) this relation is given in coordinates by

LN^{3} := ∂_{i}(\sqrt{g}g^{ij}a_{jk}g^{kl}∂_{l}N^{3}) + \sqrt{g}( tr (A_{F} S^{2}) − ⟨∇_{ℝ^{3}}H_{F} (X), N⟩)N^{3} ≤ 0 ,

which we regard as a linear elliptic equation for N^{3} with the differential operator L associated to the second variation formula (4.23). Here, g = det(g_{ij}), a_{jk} = ⟨F_{zz}(N)∂_{j}X, ∂_{k}X⟩ and the remaining coefficients are of class C^{0,α}(\overline{B}), and the leading coefficients of L are symmetric. Since X is stable we have δ^{2}F^{0}(X, Ξ) ≥ 0; hence the first eigenvalue of L is non-negative. Thus Lemma 5.1 is applicable and we have N^{3} > 0 on B.

Since X : ∂B → Γ is a topological mapping, we can apply Sauvigny’s reasoning involving degree theory as in [21, pp. 53,54] to conclude the proof. □

Remark. We have seen in (5.3) and (5.4) that there are no branch points on the boundary by the simple Hopf maximum principle argument, which is applicable because of our regularity assumptions up to the boundary. Consequently, it would suffice to assume that X is conformal and has no interior branch points and maps the boundary ∂B only weakly monotonically onto Γ, but at this point it is an open question if one can relax the smoothness assumptions to X ∈ C^{0}(\overline{B}, ℝ^{3}) ∩ C^{3}(B, ℝ^{3}).

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Received Received date / Revised version date

Fachbereich Mathematik Universität Duisburg Lotharstraße 65 47047 Duisburg E-mail address: clarenz@math.uni-duisburg.de http://numerik.math.uni-duisburg.de/people/clarenz

Mathematisches Institut Universität Bonn Beringstraße 1 53115 Bonn E-mail address: heiko@math.uni-bonn.de http://www.math.uni-bonn.de/people/heiko

The second author was supported by the Max-Planck Institute for Mathematics in the Sciences Leipzig and the Sonderforschungsbereich 611 at the University of Bonn.