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It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by this question, I'd like to ask whether a similar rigidity holds for centro-affine curvature.

More precisely, let $M\subset\mathbb R^3$ be a smooth closed convex surface (i.e., the boundary of a convex body) enclosing the origin. Its centro-affine curvature at a point $p\in M$ can be defined as $$ K(p)\cdot\langle p,\nu(p)\rangle^{-4} $$ where $K(p)$ is the Gauss curvature and $\nu(p)$ is the outer normal vector at $p$. (More generally, for a hypersurface in $\mathbb R^n$ it is $K(p)\cdot\langle p,\nu(p)\rangle^{-(n+1)}$.)

The nice thing about centro-affine curvature is that it is invariant under volume-preserving linear transformations. In particular, it is constant if $M$ is an ellipsoid centered at the origin (because such ellipsoids are equivalent to spheres). Is the converse true? In other words, is it true that every closed convex surface with constant centro-affine curvature is an ellipsoid?

Remark. For curves in $\mathbb R^2$, the condition that the centro-affine curvature is constant boils down to a second-order ODE whose solutions are ellipses and only ellipses. In $\mathbb R^3$, it is a PDE that seems to have many more solutions locally (just like in the case of constant Gauss curvature). So, if there is rigidity, it should be global only.

[EDIT] Another possible definition of centro-affine curvature is what the Legendre transform (the natural bijection between a convex body and its polar) does to the volume locally. I'm adding 'convexity' tag because some extremal properties of ellipsoids might be relevant here.

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Sorry, I don't understand what you mean by $K(p)\cdot\langle p,\nu(p)\rangle^{-4}$ . Is this correct that if a surface $S$ in $R^3$ is given as a graph of a function $z=f(x,y)$, then the centro-affine curvature is the determinant of the Hessian of $f$? –  Dmitri Apr 14 '11 at 0:03
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Dmitri, the centro-affine curvature depends on the position of the origin inside the convex body and is not invariant under transations. If I get a chance, I'll post my way of describing it. –  Deane Yang Apr 14 '11 at 2:46
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3 Answers 3

up vote 8 down vote accepted

This is a classical question that was probably posed by Blaschke and partially solved by Jorgens and Calabi but I think was finally solved completely by Cheng and Yau in "Complete Affine Hypersurfaces. Part I. The Completeness of Affine Metrics" in CPAM 1986, if you assume sufficient regularity of the boundary. These bodies can be defined without any assumption on regularity except for convexity, but I am less sure about what is known in that case.

Description of centro-affine curvature: Given a convex body with the origin in the interior, there is a uniquely defined convex function $\rho$ homogeneous of degree $2$ that is equal to $1$ along the boundary of the body. If the body is origin-symmetric, then it is the unit ball of a Banach norm and $\rho$ is just the norm squared. The Hessian of $\rho$ is homogeneous of degree $0$. The determinant of the Hessian is the centro-affine curvature.

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Here is a bit more detail on the cited paper: Communications on Pure and Applied Mathematics, Volume 39, Issue 6, pages 839–866, November 1986. onlinelibrary.wiley.com/doi/10.1002/cpa.3160390606/abstract –  Joseph O'Rourke Apr 14 '11 at 11:50
    
I'm a little hazy about this, but I am pretty sure that regularity theorems proved by Caffarelli and his collaborators imply the theorem assuming only convexity and using the appropriate definition of centro-affine curvature (as a measure). –  Deane Yang Apr 14 '11 at 19:55
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You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvature surfaces means the ellptic affine spheres. Let M be a surface with constant centro-affine curvature 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is (with center at the orgin) $$\det(D^2 u)=(u)^{-4}.$$ The answer can be found in this book, due to Blaschke, Deicke, Calabi.

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"The nice thing about centro-affine curvature is that it is invariant under volume-preserving linear transformations. In particular, it is constant if M is an ellipsoid centered at the origin (because such ellipsoids are equivalent to spheres). Is the converse true? In other words, is it true that every closed convex surface with constant centro-affine curvature is an ellipsoid?"

YES, it's true ! An elegant short proof may be found in the following paper:

Leichtweiss, Kurt On a problem of W. J. Firey in connection with the characterization of spheres. Festschrift for Hans Vogler on the occasion of his 60th birthday. Math. Pannon. 6 (1995), no. 1, 67–75.

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