On the problem of realization of a given Gaussian curvature function

V. Arnold^{1}

I. Introduction.

The Gaussian curvature of a smooth surface embedded into the Euclidian space is a smooth function on the surface. We investigate below the local realization problem : given a germ of a smooth function of two variables, find a surface whose Gaussian curvature is the given function.

It is well known that any function germ g(x, y) is realizable as the Gaussian curvature of the surface z = f(x, y), if the curvature value at the central point is not vanishing. It is also known, that it is realizable (in the same sense), if the curvature is vanishing at the central point, but its differential does not vanish. In this case, the parabolic curve is smooth.

In the case of a singular parabolic curve, the problem is more difficult. We shall see, that any parabolic curve singularity occurs for a suitable surface :

\underline{Theorem 1}. For any smooth function of two variables vanishing at its critical point of finite multiplicity, there exists a smooth surface in the Euclidian 3-space, whose Gaussian curvature coincides with the given function at a neighbourhood of the given point, (provided that the surface is identified with the plane by a suitable local diffeomorphism, depending on the function).

I recall that the multiplicity μ of a critical point 0 of function f is the dimension of the quotient vector-space of the space of formal power series in (x, y) modulo the ideal generated by the partial derivatives of the function :

μ = dim_{IR} IR[[x, y]] /{a(x, y)f_{x} + b(x, y)f_{y}} .

1Partially supported by the Russian Basic Research Foundation, project 96-01-01104

Finite multiplicity critical points of holomorphic functions are just complex isolated critical points. The codimension of the set of the functions having a critical point of infinite multiplicity is infinite. Thus typical finitedimensional families of functions contain no function having a critical point of infinite multiplicity.

\underline{Corollary 1}. Any singularity of finite multiplity μ of a plane curve is realizable as a singularity of the parabolic curve on a surface in the Euclidian 3-space (up to a diffeomorphism).

\underline{Remark 1}. The Theorem and the Corollary hold for analytical functions and diffeomorphisms as well as for the holomorphic ones or for the infinitely smooth ones and even for the finitely smooth ones (the number of the derivatives in the last case grows with μ).

\underline{Remark 2}. I have no analytical (holomorphic, smooth) counterexamples to the statement of Theorem 1 for infinite multiplicity germs.

In the formal series setting, one can prove more :

\underline{Theorem 2}. For any formal power series g(x, y) = ax + by + ... there exists a formal power series f(x, y) such that the formal power series of the Gaussian curvature of the formal surface z = f(x, y) coincides with g.

In Theorem 2, the identification of the surface with the plane (x, y) is fixed by the choice of the coordinates (x, y, z) in the 3-space.

\underline{Corollary 2}. Any formal singularity of a plane curve g(x, y) = 0 is realizable as the singularity of the parabolic curve of some formal surface z = f(x, y).

\underline{Remark 3}. I have no counterexamples (even for μ = ∞) neither in the analytical nor in the holomorphic nor in the smooth case (neither for Theorem 2 nor for its Corollary), but I guess that the above series might diverge even for finite μ.

Theorem 1 follows from Theorem 2, as it is explained below (in paragraph 3). The same arguments (in a simpler form) prove the following results.

\underline{Theorem 3}. For any function g(x, y) vanishing at a critical point of finite multiplicity, there exists a function f(x, y), whose Hessien

h = h[f] = f_{xx}f_{yy} − (f_{xy})^{2}

coindices with g at a neighbourhood of the point up to a suitable smooth change of variables :

g(x, y) = h (X(x, y), Y (x, y)) .

\underline{Remark 4}. Corollary 1 follows already from Theorem 3 (which is a simplified version of Theorem 1) since the Hessien vanishes exactly along the parabolic curve.

\underline{Theorem 4}. For any formal series g(x, y) = ax + by + ..., there exists a formal series f whose Hessien is g = h[f].

\underline{Remark 5}. Corollary 2 follows already from Theorem 4 (which is a simplified version of Theorem 2).

\underline{Remark 6}. The proofs of these results provide in fact more : one might replace in Theorems 3 and 4 the Hessien h = h[f] by any function (series) of the form, say,

g[f] = hM(x, y, f_{x}, f_{y}, f, h), M(0) ≠ 0

provided that f(0) = f_{x}(0) = f_{y}(0) = 0 (there exist many other possibilities).

This Remark implies Theorems 1 and 2, since the Gaussian curvature admits such a representation (in this case M = M(f_{x}, f_{y}), see paragraph 4 below).

\underline{Remark 7}. Theorems 1-4 have higher dimensional versions (describing the Hessiens of the functions of n variables and the Jacobians of the Gaussian mappings from the hypersurfaces in IR^{n+1} to the n-sphere).

\underline{Remark 8}. Whether the statements of Theorems 3 and 4 hold for infinite mutiplicity critical points and for analytical (holomorphic, smooth) functions f and g respectively is not known to me (see the Remarks 1 and 2 above).

The extension of the results of this paper to the (formally more general) case of the Jacobians of the Lagrangian mappings in symplectic geometry is automatical (since Lagrangian germs have gradient representatives).

2. Proof of Theorem 4.

We shall look for the series

(∗) f = f_{2} + f_{3} + ... (f_{i} = terms of degree i) ,

starting from f_{2} = y^{2} /2. This choice of f_{2} is the crucial point of the present paper : the remaining part of the proof consists of standard calculations.

The terms of the Hessien h[f] = h_{1} + h_{2} + ... are given by the obvious formulae

h_{1} = f_{3,xx}, h_{2} = f_{4,xx} + (f_{3,xx}f_{3,yy} − f_{3,xy}^{2}) ,

of the form h_{k} = f_{k+2,xx} + P_{k}([f_{m}], 3 ≤ m < k + 2) , (∗∗)

where p_{k} is a (quadratic) polynomial with respect to the second derivatives (f_{m,xx}, f_{m,xy}, f_{m,yy}) of homogeneous functions f_{m}.

The system of the equations for the functions f_{i} that we get is triangular:

h_{1}[f_{3}] = g_{1}, h_{2}[f_{3}, f_{4}] = g_{2}, ...,
h_{k}[f_{3}, ..., f_{k+2}] = g_{k}, ... .

It remains to solve the equation

f_{k+2,xx} = g_{k} − P_{k}([f_{3}], ..., [f_{k+1}]) (∗ ∗ ∗)

(whose right hand side is known) with respect to f_{k+2} = ∑ a_{p,q}x^{p}y^{q}, p + q = k +2. Denoting g_{k} −P_{k} = ∑ b_{p,q}x^{p}y^{q}, p+q = k, we solve the equation (∗∗∗), taking

a_{p,q} = b_{p−2,q} /p(p − 1) if p ≥ 2 ,

a_{0,q} = a_{1,q} = 0. Theorem 4 is proved.

\underline{Remark 9}. The extension of Theorem 4, mentioned in Remark 6, is proved by the same reasoning. Indeed, the multiplication of the Hessien by the multiplier M = c + M_{1} + M_{2} + ..., c ≠ 0, preserves the triangular character of the system h[f]M[f, h] = g for f.

To study the n-dimensional hypersurfaces and functions of n variables (Remark 7) one should start from f_{2} = ±y_{1}^{2}/2 ± ... ± y_{n−1}^{2}/2.

3. Proofs of Theorems 3 and 1

We deduce Theorem 3 from Theorem 4.

Let function g has at the origin the critical value 0 of finite multiplicity μ. For any N the proof of Theorem 4 (in paragraph 2 above) provides a polynomial f such that h[f] = g + r, where the remainder r has at the origin a zero of order at least N.

If N ≥ μ + 2, the function g + r is reducible to g by a smooth (analytical, holomorphic...) change of variables (see e.g., [1]). This proves Theorem 3.

Theorem 1 follows similarly from Theorem 2 and from the finite (μ + 2) determinacy property of the finite multiplicity critical point (used in the proof of Theorem 3 above).

4. Proof of Theorem 2.

We start from an obvious remark.

\underline{Lemma}. The Gaussian curvature g of the surface, defined in some Cartesian orthonormal coordinates (x, y, z) in the Euclidean space as the graph of a function z = f(x, y), is related to the Hessien h of the function f by the formula

g = h[f] /E^{4}[f] ,

where E^{2} = 1 + (f_{x})^{2} + (f_{y})^{2} .

\underline{Proof}. Function E is the lenght of the normal vector n = (−f_{x}, −f_{y}, 1) of the surface. The area element dS of the surface in inclined to the horizontal direction at the same angle as the angle of deviation of the normal from the vertical direction. Hence dS = Edx ∧ dy.

The Gaussian mapping G from the surface to the sphere is the product of the gradent mapping F (sending (x, y) to (u = −f_{x}, v = −f_{y}) from the surface to the horizontal plane {u, v, 1} (tangent to the unite sphere at (0, 0, 1)), followed by the central projection P from the horizontal plane to the sphere: G = P ∘ F.

The aera element of the sphere, dω, induces on the horizontal plane the form P^{∗}dω = E^{−3}du ∧ dv (the factor E^{−2} is the contribution of the fact that the point of the plane is E-times farther from the origin than the point of the sphere, and an additional factor E^{−1} is provided by the inclination of the horizontal plane to the radial direction).

By the definition of the Gaussian curvature we get

g = \frac{G^{∗}dω}{dS} = \frac{J^{∗}(E^{−3}du ∧ dv)}{Edx ∧ dy} = E^{−4}h ,

since the Hessien of f is the Jacobian of its gradient mapping F : F^{∗}du ∧ dv = h[f]dx ∧ dy. The Lemma is thus proved.

To prove Theorem 2, we look for the series (∗) as in paragraph 2. We get

E^{−4}[f] = 1 + e_{2}[f_{2}] + e_{3}([f_{2}], [f_{3}]) + ...,

e_{k} being the homogeneous part of degree k of the power series E^{−4} in (x, y).

The form e_{k} is a weighted homogeneous polynomial in the first derivatives of the functions f_{2}, ..., f_{k} (higher order terms, starting from f_{k+1}, contribute noting to e_{k}) :

e_{2}[f_{2}] = 2y^{2}, e_{3}([f_{2}], [f_{3}]) = −4yf_{3,y}, ... ,
e_{k} = e_{k}([f_{2}], ..., [f_{k}]) .

It follows that the system

h[f] E^{−4} [f] = g

of the equations for the unknowns f_{k}, (which we should solve to prove Theorem 2, conformally to the Lemma) is triangular.

Indeed, equating the homogeneous terms in the product

(h_{1} + h_{2} + ...) (1 + e_{2} + ...) = g_{1} + g_{2} + ...,

we get for the unknowns f_{k} the sequence of equations

h_{1} = g_{1}, h_{2} = g_{2}, h_{3} + h_{1}e_{2} = g_{3}, ...,

h_{k} + Q_{k} ([h_{<k}], [e_{<k}]) = g_{k} .

Substituing the expression (∗∗) from the proof of Theorem 4 to h_{k}, we get for the unknown functions f_{k} the system of equations of the form

f_{k+2,xx} = −R_{k}([f_{<k+2}]) + g_{k} (k ≥ 1)

(with known R_{k} and g_{k}). We solve these equations the same way as the equations (∗ ∗ ∗) were solved in paragraph 2, finding successively f_{3}, f_{4}, .... Theorem 2 is thus proved.

As it was explained in paragraph 3, Theorem 1 follows.

\underline{5. Parabolic curves at a flattening point}.

We call the flattening points of the surface z = f(x, y) the points where the second differential of f vanishes. In suitable (cartesian orthonormal) coordinates the Taylor series of f at a flattening point takes the form

f = f_{3} + f_{4} + .... (f_{0} = f_{1} = f_{2} = 0) .

A typical example is the generalized monkey saddle D_{4}^{±} : f = 3x^{2}y ± y^{3}. The flattening points are critical points of the Gaussian curvature, where the critical value vanishes.

\underline{Theorem 5}. The second differential of the Gaussian curature at a flattening point can’t be a nonzero nonnegative quadratic form (in particular, the flattening point can’t be a nondegenerate minimum point of the Gaussian curvature), while all the other quadratic forms are realizable as the second differentials of the Gaussian curvature at the flattening points.

\underline{Proof}. Let f_{3} = ax^{3} + 3bx^{2}y + 3cxy^{2} + dy^{3}. The Hessien of this function is

h_{2} = 36(Ax^{2} + Bxy + Cy^{2}) ,

where A = ac − b^{2}, B = ad − bc, C = bd − c^{2}.

Choose the (cartesian orthonormal) coordinates reducing the quadratic form h_{2} to the normal form (B = 0). In these coordinates ac = A + b^{2}, bd = C + c^{2}, ad = bc. Hence we have

(A + b^{2})(C + c^{2}) = b^{2}c^{2}, AC + Ac^{2} + Cb^{2} = 0 .

If the form Ax^{2} + Cy^{2} is positive definite (A > 0, C > 0) the last equality is impossible. The positive semidefinite case (A > 0, C = 0) is impossible too, since then c = 0 and hence A + b^{2} = 0.

If one of the coefficients, say A, is negative, the required realization is provided by

b = ±\sqrt{−A} , a = c = 0, d = C /b .

Theorem 5 is thus proved.

\underline{Remark 10}. Our formulae imply that the degenerate forms h_{2} are realized only by degenerate forms f_{3} = ax^{3} + 3bx^{2}y, and that the zero-form h_{2} is realized only by the bidegenerate forms f_{3} = ax^{3}.

\underline{Theorem 6}. The parabolic curve singularity at a flattening point of a surface can’t be diffeomorphic to the E_{6} singularity (to the singularity of the curve x^{3} + y^{4} at the origin).

\underline{Proof}. The equation of the parabolic curve of the surface z = f(x, y) is h[f](x, y) = 0. Choose a coordinate system for which f = f_{3} + f_{4} + ... . We should have h_{2} = 0 to realize the E_{6} singularity of h. From the Remark 10 above, we get (in suitable coordinates) f_{3} = ax^{3}. We can normalize a to be 1/6 (a ≠ 0, since for E_{6} h_{3} ∼ x^{3} ≠ 0).

Now compute h_{3}. Since we should get the E_{6} singularity for h[f], it has the form h_{3} = (px + qy)^{3}. But for f_{2} = 0, f_{3} = x^{3} /6 we get h_{3} = xf_{4,yy}. Hence q = 0, h_{3} = P x^{3}, P ≠ 0. Now we get f_{4,yy} = P x^{2}, f_{4} = x^{2}e_{2}(x, y),

h_{4} = xf_{5,yy} + f_{4,xx} f_{4,yy} − (f_{4,xy})^{2} .

Since f_{4} is divisible by x^{2}, f_{4,yy} and f_{4,xy} are divisible by x, and thus h_{4} is divisible by x.

Hence the sum of the terms of degrees 3 and 4 of the Taylor series of h[f] is equal to

H(x, y) = h_{3} + h_{4} = P x^{3} + xR_{3}(x, y) .

We shall prove now that such a function can’t have an E_{6} singularity (whose 4-th degree Taylor polynomial should be equal X^{3} + Y^{4} for some local coordinates (X, Y), defined by (x = αX + βY + ..., y = γX + δY = ...)).

Denote the expression of H in new coordinates by K(X, Y) = H(x(X, Y), y(X, Y)). Equating the terms of degree 3 to X^{3}, we get

P (αX + βy)^{3} = X^{3} .

Thus β = 0, x = αX + (higher order terms).

It follows that all the terms of degree 4 in the series K are divisible by X. Indeed, the terms of degree 4, coming from P x^{3}, have the form

T_{4} = (3P x^{2}(x − αX))_{4} = 3P (x^{2})_{2}(x − αX)_{2} .

The quadratic part of the expansion of x^{2} in the power series with argument X, Y is equal to α^{2}X^{2} since β = 0. Hence the polynomial T_{4} is divisible by X^{2}.

The fourth degree terms coming from xR_{3} do contain X for the same reason (x = αX + higher order terms). Hence the 4-th degree Taylor polynomial of K is divisible by X. Thus it can’t be equal to X^{3} + Y^{4}. Theorem 6 is proved.

\underline{Remark 11}. I do not know which curve singularities are realizable as the parabolic curve singularity at a flattening point. This question (together with similar questions on the singularities of the Hessien of the Gaussian curvature and of the Jacobian of a Lagrangian mapping) might be interesting also for higher flattenings.

References

[1] V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps, \underline{Vol. 1}, Nauka, Moscow, 1982 (English translation : Birkhäuser, 1985).

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