# Gauss–Kronecker curvature and Gauss curvature

Picture below is from Tso, Kaising, Deforming a hypersurface by its Gauss-Kronecker curvature, Commun. Pure Appl. Math. 38, 867-882 (1985). ZBL0612.53005.

On Wiki, the Gauss curvature of surface is product of principal curvature. But there is not explain about the high dimension surface. I guess the Gauss curvature of high dimension also is product of all principal curvature. But in picture below, seemly, it is not. And from
Andrews, Ben, Gauss curvature flow: The fate of the rolling stones, Invent. Math. 138, No.1, 151-161 (1999). ZBL0936.35080. I found the 2-dim Gauss curvature equal to the (0.4) below. How to get it ? Besides, what is principal minor and principal radii of curvature?

The eigenvalues of the Hessian $(\partial^2H/\partial x_i\partial x_j)(x)$, $i,j=1,\dots,n+1$, consists of zero (due to homogeneity of degree 1) and the principal radii of curvature at $p(x)$. In particular, the reciprocal of the Gauss–Kronecker curvature of $X$ at $p$ is given by the sum of the principal minor of the Hessian. If we take a set $e_1,\dots,e_n$ of orthonormal frame fields on $S^n$, then the Gauss–Kronecker curvature at $p$ (or at $x$ when regarded as a function of its unit normal) is given by $$K(x)=[\det(H_{\alpha\beta}(x)+H(x)\delta_{\alpha\beta})]^{-1},\tag{0.4}$$ where $H_{\alpha\beta}$, $\alpha,\beta=1,\cdots,n$ is the Hessian of $H$ with respect to $e_1,\dots,e_n$.

• Ricci flow, then mean curvature flow, and now you are in Gauss curvature flow.... – user99914 May 5 '17 at 4:48
• @JohnMa Because Ricci flow is too hard and I feel the mean curvature flow is not so hard , so I read the Huisken's volume preserving mean curvature flow. But there is not singularity this paper, so after read it, I turn to Andrews' Gauss curvature flow. This paper has something about rescaling the singularity. And I feel the mean curvature flow and Gauss curvature flow can be unify by k-curvature flow (although I can't found paper talk about this). So I read it. In fact, I am very confused, don't know what should to do. – lanse7pty May 5 '17 at 7:51
• There are some recent advance made by Kyeongsu Choi on Gauss curvature flow you might want to check it out. However, I do not have the big picture concerning Gauss curvature flow and has no idea where it is heading to. – user99914 May 5 '17 at 8:02

You need to give more context (i.e., early paragraphs) for this quote from the article — I'm guessing this is a hypersurface in projective space, but I'm not sure. Gauss-Kronecker curvature is something that's usually defined for submanifolds whose dimension is less than $n-1$ (ambient space $\Bbb R^n$).