Local realization of prescribed Gaussian curvature and Hessian germs: infinite-multiplicity critical points and parabolic-curve singularities at flattening points

V. I. Arnold's 1998 paper 'On the problem of realization of a given Gaussian curvature function' (Topological Methods in Nonlinear Analysis 11, no. 2, 199-206) proved that a germ of a smooth function of two variables with a critical point of finite multiplicity at the base point is locally realizable, up to a smooth local change of coordinates, both as the Gaussian curvature of a smooth surface in Euclidean 3-space and as the Hessian determinant of a smooth potential; consequently every plane-curve germ of finite multiplicity occurs as a parabolic-curve germ of some smooth surface. The paper explicitly leaves two issues open, which together form the completion of this local realization theory: (1) whether the existence theorems extend to critical points that are not of finite multiplicity, i.e. flat germs and germs whose critical locus is not isolated; and (2) which plane-curve singularity types can occur as parabolic-curve singularities at flattening points, where the paper proves prohibitions (a signature constraint on the quadratic part of the curvature and exclusion of the $E_6$ type) but no classification. The problem remains open.

Background

For a smooth surface in Euclidean 3-space written locally as a graph $z=f(x,y)$, the Gaussian curvature $K$ - the product of the two principal curvatures, i.e. of the extremal curvatures of normal sections of the surface at the point - is

$$K = \frac{f_{xx}f_{yy}-f_{xy}^{2}}{\left(1+f_x^{2}+f_y^{2}\right)^{2}}.$$

The numerator $h[f]=f_{xx}f_{yy}-f_{xy}^{2}$ is the Hessian determinant of $f$. A natural local question, studied classically and posed sharply by V. I. Arnold in his 1997 remarks on the topology of parabolic curves on surfaces and in the 1998 paper, is the realization problem: given a germ $g$ of a $C^{\infty}$ function of two variables at the origin (a germ being a function defined on some neighborhood of the point, considered up to restriction to smaller neighborhoods), is there a smooth surface, locally a graph $z=f(x,y)$ up to a smooth local reparametrization of the parameter plane, whose Gaussian curvature equals $g$ near the origin? The parallel question for the Hessian determinant asks for a smooth $f$ with $h[f]$ equal to $g$ up to a smooth local change of coordinates, and the question for the parabolic curve - the locus of points where $K=0$, which for a graph is the zero set of $h[f]$ on the parameter plane - asks which plane-curve germs occur as parabolic-curve germs.

Two cases were classically known to be solvable: $g(0)\neq 0$, and $g(0)=0$ with $dg(0)\neq 0$ (so the zero locus of $g$ is a smooth curve). The hard case is a curvature function that vanishes at a critical point. Say that the critical point of $g$ at the origin has finite multiplicity $\mu$ if the quotient algebra $\mathbb{R}[[x,y]]/(g_x,g_y)$ of the formal power series ring by the ideal generated by the two partial derivatives is finite-dimensional over $\mathbb{R}$; its dimension is $\mu$, and finiteness holds exactly when the critical point is isolated. The opposite regime - critical points that are not of finite multiplicity - contains two different phenomena: germs whose critical locus is a curve rather than a point (the local algebra is infinite-dimensional), and flat germs, whose Taylor series vanishes identically so that no finite jet carries any information.

Arnold's 1998 paper solved the finite-multiplicity case. The proof works in formal power series: starting from a quadratic term $f_2=y^{2}/2$ and writing $f=\sum_{i\ge 2}f_i$ with $f_i$ homogeneous of degree $i$, the equation $h[f]=g$ becomes a triangular system that determines the coefficients of $f_{k+2}$ from the previously computed homogeneous pieces and the prescribed $g_k$, coefficient by coefficient; the identity $g=h[f]/E^{4}[f]$ with $E^{2}=1+f_x^{2}+f_y^{2}$ shows that prescribing Gaussian curvature is the same problem with the Hessian multiplied by a unit (nonzero constant term) factor, which preserves triangularity. The formal solution is then upgraded to a smooth one by the classical finite-determinacy theorem of singularity theory: a germ with an isolated critical point of multiplicity $\mu$ is determined, up to smooth local change of coordinates, by its Taylor polynomial of order about $\mu+2$, so an approximant agreeing with the target to that order is equivalent to it. The conclusions are: every smooth germ with a critical point of finite multiplicity is realizable as a Hessian determinant and as a Gaussian curvature up to smooth local diffeomorphism, and every plane-curve germ of finite multiplicity occurs as a parabolic-curve singularity of some smooth surface.

The same paper began the analysis of the degenerate locus where realizability is constrained. A flattening point is a point of the graph where the second differential of $f$ vanishes, so that, writing the surface locally as the graph $z=f(x,y)$ (the Monge form), the Taylor series of $f$ starts with a cubic $f_3=ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}$. There the quadratic part of the Hessian - and hence of the Gaussian curvature, since the unit factor $E^{-4}$ has constant term 1 - equals $36(Ax^{2}+Bxy+Cy^{2})$ with $A=ac-b^{2}$, $B=ad-bc$, $C=bd-c^{2}$; this coefficient triple is constrained (some quadratic signatures, in particular the positive-definite one, do not arise, so a flattening point cannot be a nondegenerate local minimum of the Gaussian curvature). In addition, the parabolic-curve singularity of type $E_6$ - the curve $X^{3}+Y^{4}=0$, one of the ADE simple plane-curve singularities of Arnold's classification (the finite list of 'simple' singularities, labeled $A_k$, $D_k$, $E_6$, $E_7$, $E_8$ and given by explicit normal forms) - is proved not to occur at a flattening point, by a degree-counting argument on the Hessian of a series with vanishing quadratic part. Away from flattening points no such prohibitions exist, by the realization theorem.

The follow-up literature developed the global algebraic side of this circle of ideas (which Arnold called Hessian topology): which configurations of real algebraic curves occur as Hessian curves of polynomials, how many connected components the parabolic curve of a polynomial graph can have, how many special parabolic points (godrons - isolated points of the parabolic curve where the unique asymptotic direction, the direction of zero normal curvature, is tangent to the parabolic curve) occur on generic graphs, and the topology of spaces of hyperbolic homogeneous polynomials (in this school's terminology, homogeneous polynomials in two variables whose Hessian determinant $f_{xx}f_{yy}-f_{xy}^{2}$ is negative at every point other than the origin). The local realization questions at the degenerate cases, however, were formulated in the 1998 paper and remain open.

Problem Statement

Complete the local realization theory for prescribed Gaussian curvature (equivalently, prescribed Hessian) on smooth surfaces by settling the two questions that Arnold's 1998 finite-multiplicity theory explicitly leaves open (Remarks 8 and 11 of that paper). The two are parts of one objective - determining exactly which germs and which parabolic-curve singularity types are locally realizable.

(1) Infinite multiplicity. Do the existence theorems extend from critical points of finite multiplicity to germs whose critical point at the origin is not of finite multiplicity, i.e. either germs whose critical locus is not isolated (the algebra $\mathbb{R}[[x,y]]/(g_x,g_y)$ is infinite-dimensional) or flat germs (vanishing Taylor series)? Precisely: for every smooth germ $g$ with $g(0)=0$ and $dg(0)=0$, do there exist a smooth $f$ and a smooth local diffeomorphism $\Phi$ of the plane such that $g=h[f]\circ\Phi$ (Hessian prescription), and likewise such that $g=(h[f]/E^{4}[f])\circ\Phi$ (Gaussian-curvature prescription, $E^{2}=1+f_x^{2}+f_y^{2}$)? Or does some infinite-multiplicity germ fail to be realizable? A single explicit non-realizable germ, with a proof of the obstruction covering all smooth candidates and all local reparametrizations, would refute the extension.

(2) Parabolic singularities at flattening points. Determine the complete list of plane-curve singularity types in the simple part of the standard classification of plane-curve singularities - the types $A_k$ with $k\ge 1$, $D_k$ with $k\ge 4$, and $E_6$, $E_7$, $E_8$, in their standard normal forms - that occur as the singularity of the parabolic curve at a flattening point of a smooth surface - a point where the second differential of $f$ vanishes - given the known constraints: the quadratic part of $h[f]$ there is $36(Ax^{2}+Bxy+Cy^{2})$ with $A=ac-b^{2}$, $B=ad-bc$, $C=bd-c^{2}$, where $ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}$ is the cubic term of $f$, which excludes some signatures (in particular the positive-definite one), and the type $E_6$ ($X^{3}+Y^{4}=0$) is excluded. Both directions are required: an explicit construction for each realizable type (a surface germ whose parabolic curve carries that type at a flattening point) and an exclusion proof for each non-realizable type, together covering all types.

References

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