ON THE GROWTH OF MINIMAL GRAPHS Allen Weitsman Abstract. We consider minimal graphs u = u(, y > 0 over unbounded domains D R with D a piecewise smooth curve on which u = 0. We give some lower growth estimates for u in terms of the geometry of D. I. Introduction. Let D be an unbounded domain in R bounded by a piecewise differentiable arc, and 0 Θ(r π be the angular measure of the set D { z = r}. In the classical potential theory of harmonic functions, an important role is played by estimates involving Θ(r. For eample, if u(z is the harmonic measure of D { z <r} with respect to D { z = r}, then [T; p.6] we have Theorem A. If z D and z <κr/, then u(z 9 κ e π κr z dρ ρθ (ρ (0 <κ<, where Θ (r =Θ(r if D does not contain the entire circle z = r, andθ (r = otherwise. Theorem A is valid without topological assumptions on D, but with our current assumption that D is a piecewise differentiable arc, we have Θ (r =Θ(r forr>r 0. The proof of Theorem A for general domains is carried out by a method of Carleman. When the domain is simply connected, as in the current setting, we may use the methods of path families [F; p.0] or the Ahlfors distortion theorem [Ne; p.97]. In this work we shall rely on the method of path families. We define the asymptotic angle of D as (. β = lim sup Θ (r. r If u>0 is defined in D, then the order of u in D is given by log u(z (. α = lim sup z z D log z. 99 Mathematics Subject Classification. AMS Subject Classification: 35J60, 53A0. Key words and phrases. Minimal surfaces.
ALLEN WEITSMAN When u>0 is harmonic in D with u =0on D, it follows easily from Theorem A that (.3 α π/β. We are interested in lower bounds for growth rates for solutions to the minimal surface equation. Precisely, we consider solutions u to ( u (.4 Lu =div =0 u in D with (.5 u>0 in D, u =0 on D. In his paper [S], Spruck proved (.3 for β π, but under stringent side conditions on the behavior of u. These are i u(z as z, z D ii K(z, u(z C/( z asz, z D, where K is the Gauss curvature. The methods of [S] are nonparametric, and the conditions i and ii make the solutions resemble harmonic functions sufficiently closely so that Carleman s method can be applied. In the present work we shall use conformal parametric methods and remove the assumptions i and ii. However, here we assume that D is a single arc, whereas in [S] there are no topological constraints. We shall take advantage of the work of V. Mikljukov [M] on the moduli of path families on minimal surfaces. We shall prove Theorem.. Let D be an unbounded domain whose boundary D is a piecewise differentiable arc, and u satisfy (.4 and (.5. If β π, then (.3 holds. The case α>(β<π is ruled out for consideration in [S] by the assumption i. It is also not covered by our Theorem.. It seems reasonable in fact to conjecture that there are no nontrivial solutions u to (.4 and (.5 with β<π. This is reinforced by Nitsche s observation [Ni; p.56] that (.4 and (.5 have no nontrivial solutions if D is contained in a sector of opening less than π. I would like to thank Professor Spruck for interesting conversations and providing a preprint of his paper [S]. II. Modulus of a path family. Let D be a simply connected unbounded domain in R. Let F denote the surface given by u(z, z = i D with ds F =(u d u u d d (u d and dσ F = u d d
3 the respective length and area elements for F. For a family Γ of curves in D we define the modulus of Γ in the metric of F by mod F Γ=inf D ρ (zdσ F, the inf being taken over all nonnegative measurable functions ρ on D satisfying inf ρ(zds F. γ Γ γ The utility of the modulus comes from the elementary observation that it is a conformal invariant (cf. [M; p.65]. We shall use estimates on the modulus for path families of curves on a surface F given by solutions u(z to the minimal surface equation over domains D as in (. and (.. III. Isothermal coordinates. Let D be an unbounded domain, bounded by a simple piecewise differentiable arc and u satisfy (. and (. in D. We introduce a comple isothermal coordinate ζ for the surface F given by u over D so that the map ζ ( (ζ, (ζ,u( (ζ, (ζ is a conformal mapping onto F. We take the parameter space as the upper half plane H = {ζ : Imζ >0} with specified positively oriented points a, b D corresponding by ( (ζ, (ζ to (0, 0, (0, respectively, and.the mapping f(ζ = (ζi (ζ is then a univalent harmonic mapping of H onto D. Path families Γ in H correspond to path families on F which project to path families f(γ in D. y conformal invariance, the modulus may be computed either in H, or with the surface metric in D. When epressed in the coordinates of H, thenu = u(ζ is harmonic. With the special conditions here that u =0on H, u reflects to a harmonic function in the entire plane; since u>0inh, itmustbethatu is of the form c Imζ for some real constant c>0. IV. The modulus estimate. In this section D is an unbounded domain whose boundary is a simple piecewise differentiable curve, and D(r, R =D {r< z <R}. Fi a and b as in 3. For sufficiently large t>0 let S(t be the component of D {z : z = t} separating a from in D. Choosing r large enough so S(r separates b from in D, and R>r,letT be the subdomain of D between S(r and S(R. Let Γ = Γ(r, R be the family of curves in T that join S(r and S(R.
4 ALLEN WEITSMAN......................................................................... S(R...................................................................................................... T........................................................................ S(r......... Theorem 4.. With the above notations, suppose that u(z (4. lim sup z z D z Then (4. mod F Γ(r, R (log R D(r,R =0. ( o( d d π z (R. Proof. Following Mikljukov, we choose the density function ρ(z =( z u (z /, for z = T D(r, R andρ(z = 0 for all the remaining values z D. Hence ( z u (z dσ F (4.3 mod F Γ (. inf γ Γ γ ( z u (z / ds F For the denominator in (4.3, let γ be a curve in Γ, γ thecurveaboveitinf,andl( γ its length. Then d ρds F = γ γ. 3
5 Parametrizing γ with respect to arc length and using the fact that 3 d ds 3 (d /ds (d /ds (d 3 /ds, 3 we have γ d 3 = l( γ 0 (d /ds (d /ds (d 3 /ds ds 3 l( γ 0 d 3 /ds l( γ ds 3 0 dt t M log( R r M where M =ma S(r ( z u (z /.Thus, (4.4 γ ( ( z u (z / ds F log R r, M To obtain a bound for the numerator, we let η(z = z arctan δ(z, δ(z =u(z z, (4.5 z u (z = z ( δ (z. Tobeginwith,wenotethat η i = ( i ui arctan δ(z z 3 z δ (z z iu, z 3 and so (4.6 Also, i= u i z ( δ (z u = arctan δ(z z 3 i= u = i= η i u i u i u i δ(z u z 3 δ (z i= u i, u u i= i u i u.
6 ALLEN WEITSMAN and thus (4.7 i= u i z ( δ (z u = u z ( δ (z z ( δ (z u. Finally, by Green s Theorem (4.8 i= η i u i d d u = η u n u dz, η i= i u i u d d with n being the outer unit normal. The middle term vanishes because of the minimal surface equation. Using (4.5 (4.7 in (4.8 we then have dσ F z u (z = ( δ (z d d u z η u n dz u arctan δ(z δ(z δ (z i= i= i u i z d d u z i u i z d d u z Since this gives i= i u i z u u u, dσ F z u (z η u n dz u z u arctan δ(z u δ(z δ (z ( δ (z d d u z d d z u d d u z.
7 Since η u n dz u z η u dz u z π, and by (4. δ(z 0, we have dσ F z u (z ( o( d d π (R. z Combining this with (4.3 and (4.4 we obtain (4.. We now use the conformal invariance of the mapping H F as described in 3 together with the estimate of Theorem 4.. With a, b, f(ζ asin 3, continuing with [M; p.67] we take r>0sothats(r separates b and in D. For t r, lets (t =f (S(t so that S (t has endpoints on H in the ζ plane. Let l(t denote the Jordan curve formed by S (t along with its reflection across H and G the annular domain between l(r and l(r. Let Γ(r, R be the family of curves separating l(r andl(r ing. Then since l(r and l(r separate 0 (= f (a and (= f (b from, the modulus (in the Euclidean metric satisfies [LV; pp.3, 56 and 6 (.0] mod Γ(r, R log(6(p, π where P = min ζ l(r ζ. Now let Γ (r, R be the curves joining l(randl(ring. ThenmodΓ (r, R =/mod Γ(r, R. This follows from conformal invariance and the fact that this is the case for a true annulus [A; pp.,3]. Therefore, (4.9 mod Γ (r, R π log(6(m(r. Let m(t = min ζ(z. z =t z D Then by (4.9, the symmetry principle [A; p.6], and conformal invariance, (4.0 mod F Γ(r, R π log(6(m(r. Thus, (4.0 taken together with Theorem 4. yields
8 ALLEN WEITSMAN Theorem 4.. With the above notation and conventions,, ( log m(r lim inf ( o( d d π π. R (log R D(r,R z V. Proof of Theorem.. We assume for some β π that (.3 fails. Then (5. u(z =O( z π/β ɛ (ɛ>0,z,z D, so (4. holds and we may apply Theorem 4.. We have (5. ( log m(r π (log R ( o( dtdθ O( D(r,R t ( log m(r R β dt ( o( O( (log R r t ( log m(r β(log R ( o( (R y (5. we have (5.3 R m(r β/πo(. With s = m(r, (5.4 ma f(ζ R. ζ =s Putting together (5., (5.3, (5.4, and using the maimum principle, we obtain (5.5 ma u(f(ζ ma u(z ζ =s ζ H z =R z D O(R π/β ɛ =O(((m(R β/πo( π/β ɛ =o( s (s. Now, our hypothesis implies that u(f(ζ is harmonic and positive in H and 0 on H. As pointed out in 3, this implies that u(f(ζ = c Imζ. y (5.5 however, we have c =0, a contradiction. References [A] L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, 966. [F] W. Fuchs, Topics in the theory of functions of one comple variable, Van Nostrand, 967. [LV] O. Lehto and K. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag,, 973. [M] V. Mikljukov, Some singularities in the behavior of solutions of equations of minimal surface type in unbounded domains, Math. USSR Sbornik 44 (983, 6-73.
9 [Ne] R. Nevanlinna, Analytic functions, Springer-Verlag, 970. [Ni] J.C.C. Nitsche, On new results in the theory of minimal surfaces, ull. Amer. Math. Soc. 7 (965, 95-70. [O] R. Osserman, A survey of minimal surfaces, Dover, 986. [S] J. Spruck, Two dimensional graphs over unbounded domains, J.Inst.Math.Jussien (00, 63-640. [T] M. Tsuji, Potential theory in modern function theory, Maruzen Co. LTD., 959. Address: Department of Mathematics Purdue University West Lafayette, IN 47907-067 Email: weitsman@purdue.edu