LINEARIZATION AND EXPLICIT SOLUTIONS OF THE MINIMAL SURFACE EQUATION A bstract

We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Ah+2h = 0 describing locally germs of minimal surfaces . Here A is the LaplaceBeltrami operator on the standard two-dimensional sphere . It explains the existente of the sum operation of minimal surfaces, introduced recently. In 4-dimensional space the equation Oh + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero .


Introduction
Recently great progress was achieved in the investigation and construction of examples of minimal surfaces in R3 [1]- [3] .The Gauss map usually plays a significant role and its singularities in a sense control topology if the surface is complete [4] .It was also noticed [5], [6] that there exists a "sum" operation Ml + M2 for two minirnal surfaces Ml, M2 .It may seem to be strange, for the usual form of the minimal surface equation is essentially nonlinear .True, given a conformal minimal map R2 D U x> M C R3 we have a linear equation áx = 0 [7] .However, the condition of conformality is nonlinear itself.
In this paper, we show that apparatus of support functions usually used in convex surface theory leads to the linear and completely integradle equation of minimal surfaces in R3 .We are able to write down an explicit formula describing locally all minimal surfaces with nonvanishing curvature which is quite different from the Weierstrass description .We hope our method will be useful in global problems, too .It automatically implies the existente of the sum operation .
The main part of this work was done during the author's visit to Lithuania in 1987.I wish to thank Professor F. Weiksa for fruitful discussions.I also wish to thank the referee for his very valuable remarks, in particular, for indicating to me that the relation of equation (6) to minimal surfaces was independently stated in [7] .

. The main equation
Let M be a smooth oriented hypersurface in RN and G : M -> SN-1 be its Gauss map.Then [7] G is a local diffeomorphism wherever the Gauss curvature of M is nonzero .From now en, we assume that this condition holds at every point of M. Then G becomes a covering over its image G(M).Let U C G(M) be a simply-connected proper domain, then G-1 (U) is a disjoint union of open Vi, i E I, and GI S. : Vi -U is a diffeomorphism which we call Gi .We supply SN-1 with the canonical metric of curvature 1 .
Definition.By support function hi : U -. R we mean Lemma 1.The function hi(n) determines G71 (n) in the following way : (2) where grad hi(n) is computed in terms of the metric of SN -1 and looked at as a vector in RN .
By the nondegeneracy condition, Gi. maps isomorphically TxM onto TG ( x)S N-1 .The latter space coincides with TM as a subspace of HN so, for some ti E R, we have x -grad hi (Gi (x)) = pCi (x) , G~1(n) = pn + grad hi(n) Taking scalar product with n and accounting (1) and (grad hi(n), n) = 0 we obtain (2) .
Lemma 2 .Let A(x) be the second fundamental operator in T,,M .
Here n = Gj(x), E is the identity operator in T,,S N-1 = Tx M and Hesshi (n) is the Hessian operator [8] on the sphere SN-1 .
ProoL Denote for a moment Fi = Gi 1 on U .By (2) we have Fi(n) _ hi(n)n+ grad hi(n).Let Z E T nSN -1 .By the definition of the Hessian operator, Hess hi(n) (Z) = (Oz grad hi) (n), where 0 is the Levi-Civita connection on the sphere .For any vector field v on the sphere we have [8] Vzv = vz + (v, Z)n, where v¿ denotes usual differentiation in RN .So by ( 2), Theorem 1. Suppose N = 3 and M is minimal.Then for any proper simply-connected domain U C G(M) and any branch hi we have (6) áh i + 2hi = 0 , where A is the Laplace-Beltrami operator on the sphere 52 .Conversely, if h is a solution of ( 6) in an open U C S2 then the formula x(n) = h(n)n+grad h(n) determines a smooth map from U to R3, which is either a constant or a conformal and minimal immersion outside a locally fcnite set of isolated singularities (branch points) .
Proof.. M is minimal iff tr A(x) = 0 everywhere .For an invertible operator A in 2-space we have tr A -1 = 1, AA so by (5), tr A(x) = 0 det is equivalent to 0 = tr (h¡ (n)E + Hess h¡ (n» = 2hi + Ahi.This proves the first statement of the theorem.Now Suppose h yields (6) .Denote F(n) = h(n)n + grad h.From the proof of the Lemma 2 we know that F, (n) = h(n)E + Hessh(n) .In particular, it rneans that F, (n) maps T, S2 to itself and is symmetric in Tn S2 .Further, by (6), tr F, (n) = 0 .Note that any symmetric operator with the zero trace in 2-space is represented by a matrix b ba ) in any orthonormal basis and is thus conformal, so F is conformal, and for n E U either rank F, (n) = 2 or F, (n) = 0. Denote by Z the set of points where F, = 0.As ( 6) is elliptic, F(n) is analytic along with h(n), so if F is non-constant, then Z is nowhere dense .Outside Z, F(n) ís a conformal ímmersion and we have just shown that TF(,)F(U) = T, S2 , so, for the Gauss map we have G( From the first part of the theorem we see that Flu-z is minimal .It follows that F is harmonic in U -Z, but F is analytic in U and Z is nowhere dense, hence F is harmonic everywhere in U. Locally in conformal coordinates (x, y) we have n E Z <=> á = áF = 0, hence Y~(n) = 0 where z = x + iy and .F is holomorphic and Re .F = F, so Z is locally finite.Theorem 2. Suppose N = 4 and M is minimal .Then for any proper simply-connected U C G (M) and any branch hi, Ahi(n) + 2hi(n) doesn't change sign in U .
We turn to applications of our result .Let M1, M2 be two' minimal surfaces in R3 such that G(M1) n G(M2) has a nonempty interior in S2.In [5] and [6] their sum M1 + M2 is defined by parametrization x(n) = Gi1 (n) + G21(n) .Let h 2 ~1) and h~2) be two branches of support functions of M1 and M2 respectively.Then by (2) we have x(n) _ h(n)n+ grad h(n).where h = hW + h~2) .Next, both h t ~1) and h~2) yield (6) which is linear, hence h yields (6), too .Theorem 1 implies thus the minimality of M1 + M2 .Moreover, given a minimal M and any Killing vector field Z in S2 we can define the derivative surface MZ by (7) h(M') = h' , which is also minimal .
Denote (hi (n)E+ Hess h i (n)) X=Y, then 11 (hi (n)E+Hess hi (n)) -1 Y11 > C-1 IIYII .Actually (hi (n)E + Hess h, (n» -' is the second fundamental operator of the catenoid Mi , as we saw in Lemma 2, therefore, its eigenvalues are ± -K(n), where K(n) is the Gaussian curvature at G -1 (n) .As it is well-known (and easy to verify) that K is decaying to zero at infinity, the above inequality is impossible.Of course, the same is true about all the singular points fgip, hence, being locally finite, the set of branch points should be finite.Next, as the metric of catenoid is complete, we have f7 ~~(hi(n)E + Hess hi (n)) y(t)jj = co for any curve y : [0, oc) -S2 such tliat lim y(t) = p.Hence the same arguments t ac show that this is true for h instead of h, and finally, V) is complete.Now consider the Enneper surface [7] e : R2 _ R3 .The composition G o e with the Gauss map coincides with the inverse stereographic projection 7r-1 : R2 _ S2 -{p}, so K :~0 and the support function h is defined in S2 _ {p} .Straightforward computations show that K --0 on E, hence we can apply the same construction to obtain Proposition 2 .For any given finite set E C S2 there exists a complete minimal surface in R3 with only a, finite number of branch points whose Gauss image omits precisely the set E.
The conjecture of Meeks [4] states that for every k > 1 there exists an embedded minimal surface homeomorphic to a compact manifold punctured in k points .The problem of Osserman [7], solved by Fujimoto [4], asks whether the statement of our Proposition 2 holds for some smooth complete minimal surface (without branch points) .
2. A rather surprising phenomenon follows from our description .Namely, if Oh + 2h = 0 in an open U C S2 , then the "Monge-Ampere" 0 = det(hE+Hessh) satisfies some second order PDE .Indeed, we know from Theorem 1 that the surface M parametrized by F(n) : n H h(n)n, + grad h(n) is minimal and its curvature at F(n) is V)-1 (n) .Let's pull back on S2 the minimal surface's metric .We will obtain g = te(n)go, because the Gauss rnap F -1 (n) is conformal (here go is the spherical metric) .To compute p(n), ,,ve note that F* ds = ±O(n) dso, where ds, dso are the arca elements on M, S2 respectively.Hence, g = 10(n)Igo .Therefore, the curvature of -V)(n)go is 0 -1 (n) (compare with Rice¡-Curbastro Theorem, [10]).This is equivalent to some PDE .
3. Suppose M is a complete minimal surface of finite total curvature .Then by the theorem of Osserman [7] the Gauss map G _ : M S2 extends to a holomorphic map G of the completion M, and M -M is finite, say M -M = {pl , . . .pz} .Let G C M be the finite set_ of branch p o i n t s _ of G, say G = {q1 . . .qk} .Then N = M -((M -M) U£) (is the finite covering of S 2 -G((M -M) U G) .)Next, the support function h(n) becomes single-valued on N and we see that every complete minimal surface of finite total curvature determines a solution of the equation Oh+ 2h = 0 in a finite covering of the standard sphere punctured in a finite number of points .
4. Consider the fiat metric g = dx2 +dy 2 -dz 2 in R2,1 .If for a surface M C R 2,1 , g¡ M is positively defined, then there exists a correctly defined Gauss map G from M to the hyperboloid S : x2 + y 2z 2 = -1.It is well-known that g 1 S is the standard hyperbolic metric.Just as before we can define a support function h(n) .Formula (2) in this case reads G~1 (n) = -hi(n) + gradhi(n) .Formula 5 becomes A (x) = (-hi(n)E + Hess hi(n))-1 and (6) becomes Ohi(n) -2hi(n) = 0 for minimal surfaces M with timelike normals.