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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 $e_1,\dots,e_n$.

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  • $\begingroup$ Ricci flow, then mean curvature flow, and now you are in Gauss curvature flow.... $\endgroup$ – user99914 May 5 '17 at 4:48
  • $\begingroup$ @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. $\endgroup$ – lanse7pty May 5 '17 at 7:51
  • $\begingroup$ 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. $\endgroup$ – user99914 May 5 '17 at 8:02
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First of all, principal curvatures (without further elaboration) make sense only for a hypersurface. But, yes, in that setting, Gaussian curvature is the product of the principal curvatures. For an abstract Riemannian manifold of even dimension, not sitting in Euclidean space, one can also define (Gaussian) curvature in an intrinsic way.

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 $\Bbb R^n$).

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