What can be said about functions of constant Hessian determinant?

Question

Let $f:\mathbb{R}^2\to \mathbb{R}$ with $\det \nabla^2f = 1.$ Let's also assume that $\nabla^2 f$ is positive-definite (which we can do WLOG by adjusting the sign of $f$).

What can we say about $f$? Some obvious properties are that it is strictly convex and its graph has positive Gaussian curvature, but this would also be true if $\det \nabla^2 f>0$. Does $f$ have any special properties by virtue of having constant (and not just positive) Hessian determinant?

Answer 1

We can say that $f$ is a quadratic polynomial (same holds in any $\mathbb R^n$, not only in $\mathbb R^2$; but then convexity of $f$ must be added as an assumption). This is sometimes called a Liouville-type theorem (as it expresses the rigidity of global solutions) or Bernstein-type theorem (note the parallel with Bernstein's theorem that global solution of the minimal surface equation are affine). The result belongs to a long line of development involving the names of Jörgens, Calabi, Pogorelov, Cheng, Yau, Caffarelli... References:

1. The Monge-Ampère Equation by C. E. Gutiérrez, Chapter 4. Carefully written, but the result is stated for $u\in C^4$ only.
2. A Liouville theorem for solutions of the Monge–Ampère equation with periodic data by L. Caffarelli and Y. Li. Proof for $u\in C^2$, references to earlier literature.
3. On the improper convex affine hyperspheres by A. V. Pogorelov, Geometriae Dedicata 1 (1972), no. 1, 33–46. Distinctive geometric style of Pogorelov's writing is refreshing.
4. Some aspects of the global geometry of entire space-like submanifolds by J. Jost and Y. L. Xin. Puts the result in the context of the aforementioned Bernstein's theorem.
5. Affine Bernstein Problems and Monge-Ampère Equations by An-Min Li, Chapter 4. Another book addressing the subject, but unfortunately the proof is more of an outline.