Harnack inequality for the negative power Gaussian curvature flow

In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypersurface.


Introduction
The Harnack estimate or Harnack inequality plays an important role in geometric flows. For the heat equation, P. Li and S.-T. Yau [6] obtained the corresponding Harnack inequality by using the parabolic maximum principle. Hamilton [4,5] proved a Harnack inequality for the Ricci flow and mean curvature flow for all dimensions. For a Harnack inequality for m-power mean curvature flow, we refer to [1], [9] and [10], where m is positive. B. Chow [3] considered a Harnack inequality for m-power Gaussian curvature flow for m > 0.
In this paper, we consider the negative power Gaussian curvature flow of a compact convex hypersurface F 0 : M n → R n+1 , Here K is the Gaussian curvature and ν denotes the outward unit normal vector field. Using the similar argument in [3], we obtain a Harnack inequality for the flow (1.1).
Theorem 1.1. Suppose that F 0 : M n → R n+1 is a compact convex hypersurface.
where the notation | · | h is defined in the next section.
When 1 n ≤ b ≤ 1, some interesting results have been derived in [7], where the author considered n = 2.

Notation and evolution equations
2.1. Notation. Suppose that F : M n → R n+1 is a hypersurface. The second fundamental form is given by where ·, · denotes the standard metric on R n+1 . The induced metric and the Gaussian curvature If α = {α i } and β = {β i } are 1-forms and s = {s ij } is a symmetric positive definite covariant 2-tensor, we use the short notation Finally, we define the Laplacian-type operator by Here ∇ denotes the Levi-Civita connection of the induced metric g on M .
Let M n be a convex hypersurface in R n+1 , α = {α i } a 1-form on M n , and φ a smooth function on M . We have the following identities (see [2] or [3]): where f : (0, +∞) → R is a smooth function depending only on the Gaussian curvature K, which satisfies f ′ > 0 everywhere in order to guarantee a short time existence. Such a type of Gaussian curvature flow is called the f -Gaussian curvature flow.
Remark 2.1. For convenience, in what follows, we write f t = f (K t ) and ∂ t = ∂ ∂t . Under the f -Gaussian curvature flow, it is easy to verify the following evolution equations (compared with Lemma 3.1 in [3] Kt is well-defined.

Harnack inequality
Motivated by the self-similar solutions in [3], we define a time-dependent tensor Taking the trace of (P t ) ij with respect to (h t ) ij , we set 3.1. Evolution equation for P t . In this subsection our task is to find the evolution equation for P t . Before doing this, we first write down some elementary formulas which will be used in our complicated and tedious computation. Since f is smooth depending only on K t , we have ∇f t = f ′ t ∇K t and Hence, (2.18) can be rewritten as Using Now we evaluate the last term: Simplifying the above and plugging into the expression of ∂ t ( t f t ), we obtain the required result.
Proof. The proof is similar to that in [3]. We observe first that On the other hand, we compute the Laplacian of We compute some elementary formulas which will be used later. Note that and Using these equations, we arrive at The Laplacian of (h t ) −1 ij with respect to (h t ) ij is given by Combining those identities, we have and we also have