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Remarks on affine evolutions

Remarks on affine evolutions Abh. Math. Sere. Univ. Hamburg 66 (1996), 355-376 Remarks on Attine Evolutions By K. LEIC~ITWELSS 1 Introduction Recently B. ANDREWS submitted a paper for publication with the title "Con- traction of convex hypersurfaces by their affine normal" (see [1]). There he investigated the effect of the affine evolution xt (t > 0) of an ovaloid x in the d-dimensional space Ra as the solution of the evolution equation cOx t ~---[- = Yt (1) (Yt = affine normal vector of xt) under the initial condition x0 = x. (2) His main result was the following Theorem 1.1. Let x 9 Sd-1 ~ Ra be a smooth, strictly convex hypersurface F. Then there exists a unique, smooth solution xt to equation (1)fulfilling (2) and representing strictly convex hypersurfaces Ft which converge to a point c c Ra in finite time T. After rescaling the Ft about the final point c to make the enclosed volume Vt constant the rescaled hypersurfaces ~v t converge to an ellipsoid in the goo-topology. In a certain sense this theorem is the analogon of the theorems of G. HUrSKEN (see [6]) resp. B. CHOW (see [4]), dealing with evolutions given by = H~t) " nt http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (13)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02940814
Publisher site
See Article on Publisher Site

Abstract

Abh. Math. Sere. Univ. Hamburg 66 (1996), 355-376 Remarks on Attine Evolutions By K. LEIC~ITWELSS 1 Introduction Recently B. ANDREWS submitted a paper for publication with the title "Con- traction of convex hypersurfaces by their affine normal" (see [1]). There he investigated the effect of the affine evolution xt (t > 0) of an ovaloid x in the d-dimensional space Ra as the solution of the evolution equation cOx t ~---[- = Yt (1) (Yt = affine normal vector of xt) under the initial condition x0 = x. (2) His main result was the following Theorem 1.1. Let x 9 Sd-1 ~ Ra be a smooth, strictly convex hypersurface F. Then there exists a unique, smooth solution xt to equation (1)fulfilling (2) and representing strictly convex hypersurfaces Ft which converge to a point c c Ra in finite time T. After rescaling the Ft about the final point c to make the enclosed volume Vt constant the rescaled hypersurfaces ~v t converge to an ellipsoid in the goo-topology. In a certain sense this theorem is the analogon of the theorems of G. HUrSKEN (see [6]) resp. B. CHOW (see [4]), dealing with evolutions given by = H~t) " nt

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 27, 2008

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