Content uploaded by Ben Andrews
Author content
All content in this area was uploaded by Ben Andrews on Aug 21, 2015
Content may be subject to copyright.
Classification of limiting shapes for isotropic curve flows
Abstract
A complete classification is given of curves in the plane which contract homothetically when evolved according to a power of their curvature. Applications are given to the limiting behaviour of the flows in various situations.
... The Gaussian surface area of K, denoted by S γn,K , is the unique Borel measure that satisfies lim t→0 γ n (K + tL) − γ n (K) t = S n−1 h L dS γn,K for each convex body L in R n . Here h L is the support function of L, see (2). A more explicit formula for S γn,K is given in (2) when p = 1. ...
... Here h L is the support function of L, see (2). A more explicit formula for S γn,K is given in (2) when p = 1. The Gaussian Minkowski problem. ...
... Note that Theorem 1.1 does not assume a priori that K is origin-symmetric. This work is inspired by the foundational contributions of Andrews [2]. Andrews' methodology has recently been extended to investigate solution multiplicity for the planar L p dual Minkowski problem, as seen in the works of Liu-Lu [49] and Li-Wan [45]. ...
In this paper, we show that if $L_p$ Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk when $p\in[0,1)$. Moreover, we investigate $C^0$ estimate of the corresponding convex bodies when the density function of their Gaussian surface area measures have the uniform upper and lower bound. We obtain convex bodies' uniform upper and lower bound when $p=0$ in asymmetric situation and $p\in(0,1)$ in symmetric situation. In fact, for $p\in(0,1)$ , there is a counterexample to claim the uniform bound does not exist in asymmetric situation.
The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second named authour and at. all recently have considered $3-$dimensional rotational embedded submanifolds in $\mathbb{E}^{5}$. They gave some basic curvature properties of this type of submaifolds. Self-similar flows emerge as a special solution to the mean curvature flow that preserves the shape of the evolving submanifold. In this article we consider self-similar submanifolds in Euclidean spaces. We obtained some results related with self-shrinking rotational submanifolds in Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and sufficient conditions for these type of submanifolds to be homothetic solitons for their mean curvature flows.
Let $K$ be a smooth, origin-symmetric, strictly convex body in ${\mathbb{R}}^{n}$. If for some $\ell \in \textrm{GL}(n,{\mathbb{R}})$, the anisotropic Riemannian metric $\frac{1}{2}D^{2} \left \Vert \cdot \right \Vert_{\ell K}^{2}$, encapsulating the curvature of $\ell K$, is comparable to the standard Euclidean metric of ${\mathbb{R}}^{n}$ up-to a factor of $\gamma> 1$, we show that $K$ satisfies the even $L^{p}$-Minkowski inequality and uniqueness in the even $L^{p}$-Minkowski problem for all $p \geq p_{\gamma }:= 1 - \frac{n+1}{\gamma }$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma < \infty $. In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.
To the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.
We consider curvature-driven evolution equations for curves in the plane, and prove that the isoperimetric ratios of the evolving curves generically approach infinity if the speed of motion is proportional to curvature to a power less than 1/3.
We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n +2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-smooth initial hypersurfaces, and develop upper and lower bounds on the speed and the curvature independent of initial conditions. Applications are given to the flow by affine normal and to the existence of non-spherical homothetically contracting solutions.
Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est lisse pour tout t, il converge vers un point quand t\T et sa forme limite quand t→T est un cercle rond, avec convergence dans norme C ∞
BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a Convex Curve Self-Similar Solutions THE CURVATURE-EIKONAL FLOW FOR CONVEX CURVES Blaschke Selection Theorem Preserving Convexity and Shrinking to a Point Gage-Hamilton Theorem The Contracting Case of the ACEF The Stationary case of the ACEF The Expanding Case of the ACEF THE CONVEX GENERALIZED CURVE SHORTENING FLOW Results from Brunn-Minkowski Theory The AGCSF for s in (1/3,1) The Affine Curve Shortening Flow Uniqueness of Self-Similar Solutions THE NON-CONVEX CURVE SHORTENING FLOW An Isoperimetric Ratio Limits of the Rescaled Flow Classification of Singularities A CLASS OF NON-CONVEX ANISOTROPIC FLOWS Decrease in Total Absolute Curvature Existence of a Limit Curve Shrinking to a Point A Whisker Lemma The Convexity Theorem EMBEDDED CLOSED GEODESICS ON SURFACES Basic Results The Limit Curve Shrinking to a Point Convergence to a Geodesic THE NON-CONVEX GENERALIZED CURVE SHORTENING FLOW Short Time Existence The Number of Convex Arcs The Limit Curve Removal of Interior Singularities The Almost Convexity Theorem BIBLIOGRAPHY
An affine invariant curve evolution process is presented in this work. The evolution studied is the affine analogue of the Euclidean curve shortening flow. Evolution equations, for both affine and Euclidean invariants, are developed. An affine version of the classical (Euclidean) isoperimetric inequality is proved. This inequality is used to show that in the case of affine evolution of convex plane curves, the affine isoperimetric ratio is a non-decreasing function of time. Convergence of this affine isoperimetric ratio to the ellipse’s value (8π 2 ), as well as convergence, in the Hausdorff metric, of the evolving curve to an ellipse, is also proved.
The existence of singular minimizers in the calculus of variations has been known at least since the time of Carathéodory. Several authors had claimed that such minimizers do not exist in the context of sub-Riemannian geometry. Recently the author gave an example where the singular extremals are minimizers and are stable under perturbation. This demonstrates that they are of central importance to sub-Riemannian geometry. We describe the example and the current state of knowledge regarding singular geodesies. Several open problems are posed.