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Register a new userVolume preserving flow by powers of $k$-th mean curvature
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We consider the flow of closed convex hypersurfaces in Euclidean space $mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.
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This paper concerns closed hypersurfaces of dimension $n(geq 2)$ in the hyperbolic space ${mathbb{H}}_{kappa}^{n+1}$ of constant sectional curvature $kappa$ evolving in direction of its normal vector, where the speed is given by a power $beta (geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $mbox{Gauss}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $beta$ and $kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${mathbb{H}}_{kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow. In the present paper, we give a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new non- collapsing result of Andrews. Some parts are also inspired by the work of Perelman. In a forthcoming paper, we will give a new construction of mean curvature flow with surgery based on the theorems established in the present paper.
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of other lower order anisotropic curvature measures with nonnegative coefficients is a scaled Wulff shape. This generalizes the classical results by Schneider [Comment. Math. Helv. textbf{54} (1979), 42--60] and by Kohlmann [Arch. Math. (Basel) textbf{70} (1998), 250--256] to the anisotropic setting. The main ingredients in the proof are the generalized anisotropic Minkowski formulas and an inequality of Heintze--Karcher type for convex bodies. In the second part, we consider the volume preserving flow of smooth closed convex hypersurfaces in the Euclidean space with speed given by a positive power $alpha $ of the $k$th anisotropic mean curvature plus a global term chosen to preserve the enclosed volume of the evolving hypersurfaces. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to the Wulff shape in the Hausdorff sense. The characterization theorem for Wulff shapes via the anisotropic curvature measures will be used crucially in the proof of the convergence result. Moreover, in the cases $k=1$, $n$ or $alphageq k$, we can further improve the Hausdorff convergence to the smooth and exponential convergence.
We prove the mean curvature flow of a spacelike graph in $(Sigma_1times Sigma_2, g_1-g_2)$ of a map $f:Sigma_1to Sigma_2$ from a closed Riemannian manifold $(Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2leq -c$, $c>0$ constant, any map $f:Sigma_1to Sigma_2$ is trivially homotopic provided $f^*g_2<rho g_1$ where $rho=min_{Sigma_1}K_1/sup_{Sigma_2}K_2^+geq 0$, in case $K_1>0$, and $rho=+infty$ in case $K_2leq 0$. This largely extends some known results for $K_i$ constant and $Sigma_2$ compact, obtained using the Riemannian structure of $Sigma_1times Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.
We consider the quermassintegral preserving flow of closed emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is emph{h-convex}, then the solution of the flow becomes strictly emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.
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