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Register a new userInitial Perturbation of the Mean Curvature Flow for closed limit shrinker
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This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we pursue the stream of ideas of cite{CM3} and get a slight refinement of their results. We apply the invariant manifold theory from hyperbolic dynamics to study the dynamics close to a closed shrinker that is not a sphere. In the second theorem, we show that if a hypersurface under the rescaled mean curvature flow converges to a closed shrinker that is not a sphere, then a generic perturbation on initial data would make the flow leave a small neighborhood of the shrinker and never come back.
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This is the second paper in the series to study the initial perturbation of mean curvature flow. We study the initial perturbation of mean curvature flow, whose first singularity is modeled by an asymptotic conical shrinker. The noncompactness of the limiting shrinker creates essential difficulties. We introduce the Feynman-Kac formula to get precise asymptotic behaviour of the linearized rescaled mean curvature equation along an orbit. We also develop the invariant cone method to the non-compact setting for the local dynamics near the shrinker. As a consequence, we prove that after a generic initial perturbation, the perturbed rescaled mean curvature flow avoids the conical singularity.
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.
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 prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci curvature in hyperbolic space.
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