gpt 6 astra ultra attempt on Math-332

Nirenberg Problem on the Two-Sphere

Definitions and mathematical statement

Let $g_0$ be the unit round metric on $S^2$, with Gaussian curvature one, and take the Laplacian convention $\Delta_{g_0}=\operatorname{div}\nabla$. A smooth conformal metric is $g=e^{2u}g_0$ with $u\in C^\infty(S^2,{\mathbb{R}})$. Its curvature is \[ K_g=e^{-2u}(1-\Delta_{g_0}u). \] The Nirenberg problem asks for necessary and sufficient conditions on $K\in C^\infty(S^2,{\mathbb{R}})$ for solvability of \[ -\Delta_{g_0}u+1=K e^{2u}\quad\text{on }S^2. \] In particular, a solution must satisfy $\int_{S^2}K e^{2u}{\,\mathrm{d}} A_{g_0}=4\pi$, but this identity involves the unknown $u$ and is not a complete characterization. The task is exact existence within the fixed conformal class, not uniqueness of $u$ or existence of a curvature function after an unrelated change of metric.

Formulation and concept references: [S1].

Short English statement

Which smooth functions on the sphere occur as Gaussian curvature after a smooth conformal change of the round metric?

Research attempt

Scope and current references.

A complete necessary-and-sufficient characterization of smooth prescribed curvature is UNRESOLVED here. The source [S1] discusses bubbling and loss of compactness. The newer reference [S2], submitted 31 March 2026, reports verified existence near particular computed approximations and determines their symmetries. Its abstract does not claim a characterization for every smooth $K$. We derive an exact obstruction, prove a local existence class, construct nonconstant examples, and show explicitly why a naive compactness argument fails even for constant positive curvature.

All integrations in this attempt use the round area form $dA_0$. The sign convention is $\Delta=\operatorname{div}\nabla$, so $\int v\Delta v=-\int|\nabla v|^2$ and the coordinate functions $x_i$ on the unit sphere satisfy $\Delta x_i=-2x_i$.

Elementary necessary conditions and constant data.

Integrating the equation gives \[ \int_{S^2}K e^{2u}\,dA_0=4\pi. \tag{328.1} \] Therefore $K$ must be positive somewhere. This excludes nonpositive functions, including zero, but it is not sufficient. If $K\equiv c>0$, the constant $u=-\tfrac12\log c$ is a solution. If $K$ is multiplied by a positive constant $a$, replacing a solution $u$ by $u-\tfrac12\log a$ supplies a solution for $aK$. This scaling changes area and keeps the equation's constant term equal to one.

At a maximum of $u$, the equation gives $K e^{2u}\geq1$ at that point. At a minimum it gives $K e^{2u}\leq1$. These are statements at extrema of the unknown $u$, not bounds on the extrema of $K$ alone. A maximum-principle argument that identifies their locations would introduce an unproved relation.

Deriving the conformal-vector-field obstruction.

Fix a coordinate function $z=x_i$, and let $X=\nabla z$. On the round sphere, \[ \operatorname{div}X=-2z,\qquad \operatorname{Hess}z=-z g_0. \tag{328.2} \] These identities follow, for example, by differentiating the restriction of a linear Euclidean coordinate to the sphere. We prove the Kazdan--Warner identity directly rather than treating it as an extra hypothesis: \[ \int_{S^2}X(K)e^{2u}\,dA_0=0. \tag{328.3} \] Differentiate $Ke^{2u}=1-\Delta u$ in the $X$ direction. The left side of $328.3$ becomes \[ -\int X(\Delta u)-2\int X(u)+2\int X(u)\Delta u. \tag{328.4} \] Integration by parts using $328.2$ gives $\int X(\Delta u)=2\int z\Delta u=-4\int zu$ and $\int X(u)=2\int zu$, so the first two terms cancel. For the last term, \[ \begin{split} \int X(u)\Delta u &=-\int\operatorname{Hess}u(X,\nabla u) -\int\operatorname{Hess}z(\nabla u,\nabla u)\\ &=-\tfrac12\int X(|\nabla u|^2)+\int z|\nabla u|^2=0. \end{split} \] The final equality again uses $\operatorname{div}X=-2z$. This proves $328.3$, including its weight $e^{2u}$. Removing that weight would give a different and generally false necessary condition.

Strictly positive curvature data can still be impossible.

Take $K=a+bz$ with $b\ne0$. Since $|\nabla z|^2=1-z^2$, \[ X(K)=b(1-z^2). \] It has a strict, fixed sign away from the two poles. Because $e^{2u}$ is everywhere positive, its integral in $328.3$ cannot vanish. Thus no smooth conformal solution exists. In particular, $a>|b|>0$ gives an everywhere positive function which is not realizable in the fixed conformal class.

This is an obstruction for the exact function at its given points. Changing coordinates by a conformal transformation acts on the curvature function by composition and also changes the conformal factor. It cannot be used to replace the desired $K$ with an unrelated function and still claim to have solved the original prescription.

A local existence theorem for antipodally even data.

Let $0<\alpha<1$ and use the Banach spaces of antipodally even real functions $C^{2,\alpha}_{\mathrm{even}}$ and $C^{0,\alpha}_{\mathrm{even}}$. Define \[ \mathcal F(u)=e^{-2u}(1-\Delta u). \] It is a smooth map between these spaces, $\mathcal F(0)=1$, and \[ D\mathcal F_0(v)=-\Delta v-2v. \tag{328.5} \] The spherical harmonic eigenvalues of $-\Delta$ are $\ell(\ell+1)$, and the eigenspace for $\ell$ has antipodal parity $(-1)^\ell$. The only zero eigenvalue of $328.5$ occurs at $\ell=1$, which is odd. Thus $328.5$ has trivial kernel on the even subspace and is an isomorphism there. More explicitly, self-adjoint elliptic Fredholm theory gives surjectivity after the kernel is removed, and the Schauder estimate gives a bounded inverse. These standard elliptic and spectral facts are the external inputs.

The Banach-space inverse-function theorem now gives a neighborhood of $1$ in $C^{0,\alpha}_{\mathrm{even}}$ in which every $K$ has a unique sufficiently small even $C^{2,\alpha}$ solution. For smooth $K$, the equation and Schauder bootstrapping make $u$ smooth. This proves a genuine local prescription result with a clearly stated symmetry and norm restriction.

On the full function space $328.5$ has its three-dimensional coordinate-function kernel. Ignoring it would make an inverse-function argument invalid. Removing that kernel by a conformal gauge requires compatibility with the nonlinear obstruction $328.3$, not merely discarding three equations. The even subspace avoids it because the symmetry excludes exactly those modes; it does not imply global solvability for all even positive functions.

A directly manufactured nonconstant family.

Put $P_2(z)=(3z^2-1)/2$, so $\Delta P_2=-6P_2$. For real $\varepsilon$ let \[ u_\varepsilon=\varepsilon P_2(z),\qquad K_\varepsilon=e^{-2\varepsilon P_2(z)} (1+6\varepsilon P_2(z)). \tag{328.6} \] Substitution proves the equation exactly. Since $-1/2\leq P_2\leq1$, the curvature is strictly positive whenever $-1/6<\varepsilon<1/3$. It is nonconstant for nonzero sufficiently small $\varepsilon$, as follows either by differentiating at zero or by comparing its values as a function of $P_2$. Moreover $\int P_2\,dA_0=0$, so $\int K_\varepsilon e^{2u_\varepsilon}\,dA_0=4\pi$ directly. The identity $328.3$ also holds: for $z=x_3$, its integrand is odd under $z\mapsto-z$, while for the other coordinate directions reflection symmetry in those coordinates gives the same cancellation.

Manufacturing $K$ from $u$ verifies a class of examples but does not invert the curvature map for arbitrary prescribed data. In particular, it supplies no criterion deciding whether an independently given $K$ belongs to the image of that map.

An explicit failure of compactness with $K=1$.

In stereographic coordinates $w\in\mathbb R^2$ the round metric is $4|dw|^2/(1+|w|^2)^2$. Pull it back under the conformal dilation $w\mapsto\lambda w$. The resulting conformal factor is \[ u_\lambda(w)=\log\lambda+\log(1+|w|^2) -\log(1+\lambda^2|w|^2). \tag{328.7} \] Each factor extends smoothly over the point at infinity, because it comes from a smooth conformal automorphism of the sphere. Pullback preserves constant Gaussian curvature, so every $u_\lambda$ solves the equation for $K=1$. Yet $u_\lambda(0)=\log\lambda\to\infty$.

The corresponding area measure is \[ e^{2u_\lambda}dA_0 =\frac{4\lambda^2}{(1+\lambda^2|w|^2)^2}\,dw. \] Its total mass is $4\pi$, while its mass outside the coordinate disk of radius $R>0$ is exactly \[ \frac{4\pi}{1+\lambda^2R^2}\longrightarrow0. \tag{328.8} \] This follows by one radial integral. Thus a fixed total curvature and positive smooth prescribed curvature do not by themselves give a uniform bound on all solutions. In this example the degeneration is generated by the conformal symmetry group. For general data, excluding or controlling concentration requires additional arguments; one cannot simply invoke elliptic compactness from $328.1$.

What the obstruction does and does not characterize.

The statement "there exists a positive weight making the integrals vanish" is weaker than finding that weight as $e^{2u}$ for a solution. The PDE links the weight to its own Laplacian, while a finite list of moment equations does not. Conversely, evaluating $328.3$ with weight one and rejecting a $K$ on that basis is not legitimate. The necessary conditions depend on the unknown metric.

Likewise local invertibility near $u=0$ on even functions cannot be continued along an arbitrary path of curvatures without control of the linearized operator and compactness. Along such a path solutions can leave the small neighborhood, develop degeneracy, or concentrate. Formula $328.7$ gives an explicit reason that a bound must be proved rather than inferred from smoothness of the right side.

Checks and outcome.

The saved diagnostic differentiates the polynomial $P_2$ through the zonal Laplacian $((1-z^2)P_2')'$, checks the coefficient in $328.6$, and numerically compares the radial mass formula $328.8$ with its antiderivative. These limited calculations do not certify existence for an arbitrary curvature function.

The proved results are the exact conformal-vector obstruction, the nonexistence of all nonconstant affine coordinate curvatures, the small even-data existence theorem, and the explicit solution families. The first general missing step is a complete criterion that controls both obstruction and concentration for arbitrary smooth data. The global characterization remains UNRESOLVED.

Sources

• S1 Struwe, Bubbling and Topological Degeneration in the Calculus of Variations $2023$. https://link.springer.com/article/10.1007/s00032-022-00370-1. Formulation source.
• S2 Status source for Nirenberg Problem on the Two-Sphere. https://arxiv.org/abs/2603.29544. Further reference; not a proof endorsement.