Gauss curvature estimates for minimal graphs

We estimate the Gauss curvature of nonparametric minimal surfaces over the two-slit plane C\((−∞,−1]∪[1,∞)) at points above the interval (−1, 1).


Introduction. Statement of results.
The relation between nonparametric minimal surfaces over simply connected domains and harmonic mappings is given by the Weierstrass Representation (see, e.g.[1]).
Theorem.A nonparametric surface X over a simply connected domain Ω = C is minimal if and only if there is a harmonic univalent and sense preserving mapping f = h + ḡ of the unit disk D onto Ω such that its dilatation is the square of an analytic function.Moreover, X can be represented as Re f (z), Im f (z), 2 Im bh dz : z ∈ D , where the dilatation ω(z) = g (z)/h (z) = b 2 (z).
If the surface is a minimal graph and has a representation given in the Theorem, then the Gauss curvature at the point that lies over w = f (z) is given by the formula (see [1] pp. 173-184) If we apply the Schwarz-Pick lemma, we get the estimates (1.1) In particular, If X is a minimal graph above the unit disk D and f is a harmonic map of D onto itself such that f (0) = 0, then the Heinz lemma gives the estimate 2) is not sharp.However, if one assumes that the minimal surface over D has a horizontal tangent plane at the point above the origin, then we have the sharp estimate Proofs and further details may be found in [1].Sharp bounds for Gauss curvature for minimal graphs over regions such as a half-plane, an infinite strip and the whole plane with a linear slit along negative real axis were found by Hengartner and Schober in [3].S. H. Jun [4] obtained some estimates for the slit plane C \ [a, b].
We will use Hengartner-Schober [3] approach for the case of the plane with two linear slits along the real axis.We mention that harmonic mappings onto the two-slit plane were studied by Livingston [5] and Grigoryan and Szapiel [2].
Let a < 0 < b, and We will be interested in the symmetric case when a = −b.
Then c 1 = −c 2 (see [5]).Moreover, c 2 is the solution of the equation Indeed, by Lemma 1 in [5], c 2 is a unique zero of the function and a calculation shows that the zero of T must satisfy (1.4).This means, in particular, that in the symmetric case c in formula (1.3) lies in the interval ).Our main results are the following theorems.
Moreover, if X has a horizontal tangent plane above the origin, then Theorem 2. Under the assumptions of Theorem 1, the Gauss curvature at the point above p ∈ (−1, 1) satisfies We also show that estimate (1.5) is sharp.

Proofs.
In the proof of Theorem 1 we will use the following lemma due to Hengartner and Schober [3].
Proof of Theorem 1.
Moreover, we can assume that λ > 0. Then u(r) = Re f (r) = f (r) is an increasing function that maps the interval

Using the equality
)h , we get from (1.1) that the Gauss curvature of the minimal graph over Ω above zero satisfies It follows from (2.1) and (2.2) that Now, using Lemma HS, we get where M is as in the Lemma HS.Since ϕ (0) = λ, it follows from estimate (2.3) that Hence To complete the proof, we use the function H defined in Lemma 5.2 by It was shown in [3] that if α = β then H assumes a maximum over D at or Since the maximum of both we get the desired estimate.If b(0) = 0, which means that the surface X has a horizontal tangent plane at the point above the origin, we have Remark.Under the assumption that the tangent plane is horizontal, inequality (1.5) is sharp.Actually, as in Livingston's paper one can consider the family F of harmonic mappings obtained by the shear construction of the functions ϕ(z) = λ z 1 + cz + z 2 for which equations (2.1) and (2.2) are satisfied.Then the family F contains all harmonic mappings f of the disk D onto Ω such that f (0) = 0, f z (0) > 0, f z (0) = 0, and F is the closure of these mappings in the topology of the uniform convergence on compact subsets of D. Taking ϕ(z) = h(z) − g(z) = λz 1+z 2 and the dilatation ω(z) = z 2 , we get from equation (2.1) (or (2.2)) that λ = 4/π and we find f ∈ F of the form This function maps the unit disk onto the vertical strip {w : | Re w| < 1}.It lifts to a nonparametric minimal surface whose Gaussian curvature at the point above the origin is π 2 /4.
Since f can be approximated uniformly on compact sets by harmonic mappings of the disk D onto Ω with dilatations ω n = r n z 2 , where 0 < r n < 1, lim n→∞ r n = 1, we obtain a sequence of minimal surfaces over Ω whose Gaussian curvature above the origin converges to π 2 /4.
If we apply the shear construction to the same ϕ, but with the dilatation ω(z) = −z 2 , we get

− 1 ,P
is analytic in D, with P (0) = 1, Re P (z) > 0 and −2 ≤ c ≤ 2.Moreover, Livingston showed that for given a, b with a + b ≥ 0 one can find numbers c 1 and The function f 1 lifts to the nonparametric minimal surface whose Gaussian curvature at the point above the origin is π 2 /16.Figures1 and 2depict the minimal surfaces above the harmonic shears f and f 1 , respectively.

Figure 1 .
Figure 1.Minimal surface over a strip