In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in C∞-topology to a smooth strictly convex soliton as t approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bod...

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be globally definable one parameter family of $C^2$-hypersurfaces $\mathbb{R}^n$. Upon defining notion generalized critical value for such we show that functions $s \to |K(s)|$ and $s\to K(s)$, respectively total absolute Gauss-Kronecker curvature $T_s$, are continuous in any neighb...

We use the entropy function formalism to study the effect of the Gauss-Bonnet term on the entropy of spherically symmetric extremal black holes in heterotic string theory in four dimensions. Surprisingly the resulting entropy and the near horizon metric, gauge field strengths and the axion-dilaton field are identical to those obtained by Cardoso et. al. for a supersymmetric version of the theor...

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform L-bounded solution sequence for p > 2, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. Th...