A remark on rotationally symmetric solutions to the centroaffine Minkowski problem
Let $M$ be a smooth closed convex hypesurface in $\mathbb{R}^{n+1}$ enclosing the origin. The centroaffine curvature of $M$ at a point $p$ is defined to be $K/d^{n+2}$ where $K$ is the Gauss curvature of $M$ and $d$ is the distance from the origin to the tangent hyperplane to $M$ at $p$. The centroaffine Minkowski problem is to find a closed convex hypersurface $M$ in $\mathbb{R}^{n+1}$ with prescribed centroaffine curvature at each point of $M$. Analytically this is equivalent to solving the Monge-Ampère equation \[ \det(\nabla^2 H + HI) = \frac{f}{H^{n+2}} \quad\text{on}\quad \mathbb{S}^n \tag{*} \] where $f$ is a given positive function, $H$ is the support function of $M$, $I$ is the identity matrix, and $\nabla^2H$ is the covariant Hessian matrix of $H$ with respect to an orthonormal frame on $\mathbb{S}^n$. This equation also arises in the $L_p$-Minkowski problem with $p=-n-1$ studied by \textit{E. Lutwak} [J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)]. It also corresponds to the critical exponent for the Blaschke-Santalo inequality studied by \textit{E. Lutwak} and \textit{G. Zhang} [J. Differ. Geom. 47, No. 1, 1--16 (1997; Zbl 0906.52003)]. The equation $(*)$ is also significant for other geometric problems. The existence theory for $(*)$ is rather complicated and incomplete. \par Here the author is concerned with the case that $f$ and $H$ are rotationally symmetric with respect to the $x_{n+1}$ axis in $\mathbb{R }^{n+1}$. Then $f$ can be regarded as a function on $[0,\pi]$ via the identification $f(\theta)= f(x_1,\dots,x_{n+1})$ with $x_{n+1}=\cos \theta$. \par In previous work the author and \textit{X.-J. Wang} [J. Differ. Equations 254, No. 3, 983--1005 (2013; Zbl 1273.52006)] proved, among other things, that $(*)$ has a solution in the rotationally symmetric case under a number of technical assumptions on $f$, one of which is that $f'(\pi/2)=0$. Building on previous work with \textit{H. Jian} [Discrete Contin. Dyn. Syst. 36, No. 2, 971--980 (2016; Zbl 1326.35169)], the author completely removes this condition in the cases $n=1,2$. Further, by a blow-up analysis he shows the existence of solutions in the radially symmetric case with $n\ge 3$ under conditions on $f$ that are somewhat different from those used previously by the author and Wang; in particular $f'(\pi/2)=0$ is not required, but if this is satisfied, the result reduces to a previous theorem of the author and Wang.
scientific article; zbMATH DE number 7005742
scientific article