Regularity Theory for Mean Curvature Flow:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Birkhäuser Boston
2004
|
| Schriftenreihe: | Progress in Nonlinear Differential Equations and Their Applications
57 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | This book started as a set of informal notes on Brakke's regularity theory for mean curvature flow ([B]). These notes focussed on the special case where smooth solutions of mean curvature flow develop singularities for the first time, thus expressing the underlying ideas almost entirely in the language of differential geometry and partial differential equations. In particular, notation from geometric measure theory was kept to a minimum. After I gave lectures on Brakke's work during 1994 in the Mathematics Departments at Stanford University and the University of Tübingen and at a workshop on Motion by Mean Curvature in Trento, I was encouraged by a number of colleagues to publish my notes. Since that time, but particularly since 1978, when Brakke's work first appeared, there have been many new developments in mean curvature flow starting with Huisken's work in 1984 ([Hu1]). Some of these have resulted in significant simplifications of Brakke's original arguments as well as improvements of his results in special situations. This includes particularly the case of evolving hypersurfaces with positive mean curvature. Remarkably though, in the general case the estimate of the singular set provided in Brakke's main regularity theorem ([B], Theorem 6.12) has not been improved upon to date. The bulk of the material in this text is based on lectures I gave in the department of Mathematics at the Universität Freiburg, Germany, from November 2000 to February 2001 and at Monash University, Melbourne, Australia during the first half of 2001. The central theme is the regularity theory for mean curvature flow leading up to a proof of Brakke's main regularity theorem ([B], Theorem 6.12) for the special case where smooth solutions develop singularities. In this self contained presentation, I have replaced many of Brakke's original techniques by more recent methods wherever this led to a clear simplification of his x Preface arguments. Some of his original ideas, in only slightly modified form, have been included in an appendix. Under additional assumptions such as symmetry conditions or dimensional restrictions on the solution or sign conditions on the mean curvature, improved estimates for the dimension of the singular set or refined descriptions of the behaviour of the solution near singularities can be established. I have, however, decided not to include a treatment of such results in this presentation. In particular, the book does not cover the following important contributions: Altschuler, Angenent and Giga's work on isolated singularities of surfaces of revolution ([AAG]), Angenent and Velazquez's construction of solutions exhibiting degenerate neck pinches ([AV]), Hamilton's influential Harnackin equality for convex solutions ([Ha4]), Huisken's classification of self-similar solutions with nonnegative mean curvature ([Hu3]), Huisken and Sinestrari's and White's asymptotic description of singularities in the mean convex case ([HS1], [W4]), Ilmanen's results on smooth blow-ups in two dimensions ([I1],[I2]) as well as White's dimension reduction argument ([W1]). The latter works without additional assumptions on the solution but so far implies improved (and optimal) estimates for the singular set only in the mean convex case ([W1],[W2],[W4]). I also have chosen not to include important alternative approaches to mean curvature flow such as the level-set approach adopted by Evans and Spruck ([ES1]-[ES3]) and by Chen, Giga and Goto ([CGG],[GG1],[GG2]) as well as the work of Ilmanen ([I1]) which establishes a link between level-set flow and Brakke's varifold solution framework |
| Beschreibung: | 1 Online-Ressource (XIII, 165 p) |
| ISBN: | 9780817682101 9780817637811 |
| DOI: | 10.1007/978-0-8176-8210-1 |
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| 245 | 1 | 0 | |a Regularity Theory for Mean Curvature Flow |c by Klaus Ecker |
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| 490 | 1 | |a Progress in Nonlinear Differential Equations and Their Applications |v 57 | |
| 500 | |a This book started as a set of informal notes on Brakke's regularity theory for mean curvature flow ([B]). These notes focussed on the special case where smooth solutions of mean curvature flow develop singularities for the first time, thus expressing the underlying ideas almost entirely in the language of differential geometry and partial differential equations. In particular, notation from geometric measure theory was kept to a minimum. After I gave lectures on Brakke's work during 1994 in the Mathematics Departments at Stanford University and the University of Tübingen and at a workshop on Motion by Mean Curvature in Trento, I was encouraged by a number of colleagues to publish my notes. Since that time, but particularly since 1978, when Brakke's work first appeared, there have been many new developments in mean curvature flow starting with Huisken's work in 1984 ([Hu1]). Some of these have resulted in significant simplifications of Brakke's original arguments as well as improvements of his results in special situations. | ||
| 500 | |a This includes particularly the case of evolving hypersurfaces with positive mean curvature. Remarkably though, in the general case the estimate of the singular set provided in Brakke's main regularity theorem ([B], Theorem 6.12) has not been improved upon to date. The bulk of the material in this text is based on lectures I gave in the department of Mathematics at the Universität Freiburg, Germany, from November 2000 to February 2001 and at Monash University, Melbourne, Australia during the first half of 2001. The central theme is the regularity theory for mean curvature flow leading up to a proof of Brakke's main regularity theorem ([B], Theorem 6.12) for the special case where smooth solutions develop singularities. In this self contained presentation, I have replaced many of Brakke's original techniques by more recent methods wherever this led to a clear simplification of his x Preface arguments. Some of his original ideas, in only slightly modified form, have been included in an appendix. | ||
| 500 | |a Under additional assumptions such as symmetry conditions or dimensional restrictions on the solution or sign conditions on the mean curvature, improved estimates for the dimension of the singular set or refined descriptions of the behaviour of the solution near singularities can be established. I have, however, decided not to include a treatment of such results in this presentation. | ||
| 500 | |a In particular, the book does not cover the following important contributions: Altschuler, Angenent and Giga's work on isolated singularities of surfaces of revolution ([AAG]), Angenent and Velazquez's construction of solutions exhibiting degenerate neck pinches ([AV]), Hamilton's influential Harnackin equality for convex solutions ([Ha4]), Huisken's classification of self-similar solutions with nonnegative mean curvature ([Hu3]), Huisken and Sinestrari's and White's asymptotic description of singularities in the mean convex case ([HS1], [W4]), Ilmanen's results on smooth blow-ups in two dimensions ([I1],[I2]) as well as White's dimension reduction argument ([W1]). The latter works without additional assumptions on the solution but so far implies improved (and optimal) estimates for the singular set only in the mean convex case ([W1],[W2],[W4]). | ||
| 500 | |a I also have chosen not to include important alternative approaches to mean curvature flow such as the level-set approach adopted by Evans and Spruck ([ES1]-[ES3]) and by Chen, Giga and Goto ([CGG],[GG1],[GG2]) as well as the work of Ilmanen ([I1]) which establishes a link between level-set flow and Brakke's varifold solution framework | ||
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Datensatz im Suchindex
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| any_adam_object | |
| author | Ecker, Klaus |
| author_facet | Ecker, Klaus |
| author_role | aut |
| author_sort | Ecker, Klaus |
| author_variant | k e ke |
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| dewey-ones | 516 - Geometry |
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-0-8176-8210-1 |
| format | Electronic eBook |
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| id | DE-604.BV042419210 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:04Z |
| institution | BVB |
| isbn | 9780817682101 9780817637811 |
| language | English |
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| series | Progress in Nonlinear Differential Equations and Their Applications |
| series2 | Progress in Nonlinear Differential Equations and Their Applications |
| spelling | Ecker, Klaus Verfasser aut Regularity Theory for Mean Curvature Flow by Klaus Ecker Boston, MA Birkhäuser Boston 2004 1 Online-Ressource (XIII, 165 p) txt rdacontent c rdamedia cr rdacarrier Progress in Nonlinear Differential Equations and Their Applications 57 This book started as a set of informal notes on Brakke's regularity theory for mean curvature flow ([B]). These notes focussed on the special case where smooth solutions of mean curvature flow develop singularities for the first time, thus expressing the underlying ideas almost entirely in the language of differential geometry and partial differential equations. In particular, notation from geometric measure theory was kept to a minimum. After I gave lectures on Brakke's work during 1994 in the Mathematics Departments at Stanford University and the University of Tübingen and at a workshop on Motion by Mean Curvature in Trento, I was encouraged by a number of colleagues to publish my notes. Since that time, but particularly since 1978, when Brakke's work first appeared, there have been many new developments in mean curvature flow starting with Huisken's work in 1984 ([Hu1]). Some of these have resulted in significant simplifications of Brakke's original arguments as well as improvements of his results in special situations. This includes particularly the case of evolving hypersurfaces with positive mean curvature. Remarkably though, in the general case the estimate of the singular set provided in Brakke's main regularity theorem ([B], Theorem 6.12) has not been improved upon to date. The bulk of the material in this text is based on lectures I gave in the department of Mathematics at the Universität Freiburg, Germany, from November 2000 to February 2001 and at Monash University, Melbourne, Australia during the first half of 2001. The central theme is the regularity theory for mean curvature flow leading up to a proof of Brakke's main regularity theorem ([B], Theorem 6.12) for the special case where smooth solutions develop singularities. In this self contained presentation, I have replaced many of Brakke's original techniques by more recent methods wherever this led to a clear simplification of his x Preface arguments. Some of his original ideas, in only slightly modified form, have been included in an appendix. Under additional assumptions such as symmetry conditions or dimensional restrictions on the solution or sign conditions on the mean curvature, improved estimates for the dimension of the singular set or refined descriptions of the behaviour of the solution near singularities can be established. I have, however, decided not to include a treatment of such results in this presentation. In particular, the book does not cover the following important contributions: Altschuler, Angenent and Giga's work on isolated singularities of surfaces of revolution ([AAG]), Angenent and Velazquez's construction of solutions exhibiting degenerate neck pinches ([AV]), Hamilton's influential Harnackin equality for convex solutions ([Ha4]), Huisken's classification of self-similar solutions with nonnegative mean curvature ([Hu3]), Huisken and Sinestrari's and White's asymptotic description of singularities in the mean convex case ([HS1], [W4]), Ilmanen's results on smooth blow-ups in two dimensions ([I1],[I2]) as well as White's dimension reduction argument ([W1]). The latter works without additional assumptions on the solution but so far implies improved (and optimal) estimates for the singular set only in the mean convex case ([W1],[W2],[W4]). I also have chosen not to include important alternative approaches to mean curvature flow such as the level-set approach adopted by Evans and Spruck ([ES1]-[ES3]) and by Chen, Giga and Goto ([CGG],[GG1],[GG2]) as well as the work of Ilmanen ([I1]) which establishes a link between level-set flow and Brakke's varifold solution framework Mathematics Differential equations, partial Global differential geometry Differential Geometry Measure and Integration Partial Differential Equations Theoretical, Mathematical and Computational Physics Mathematik Fluss Mathematik (DE-588)4489499-5 gnd rswk-swf Mittlere Krümmung (DE-588)4235959-4 gnd rswk-swf Regularität (DE-588)4049074-9 gnd rswk-swf Fluss Mathematik (DE-588)4489499-5 s Mittlere Krümmung (DE-588)4235959-4 s Regularität (DE-588)4049074-9 s 1\p DE-604 Progress in Nonlinear Differential Equations and Their Applications 57 (DE-604)BV036582883 57 https://doi.org/10.1007/978-0-8176-8210-1 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ecker, Klaus Regularity Theory for Mean Curvature Flow Progress in Nonlinear Differential Equations and Their Applications Mathematics Differential equations, partial Global differential geometry Differential Geometry Measure and Integration Partial Differential Equations Theoretical, Mathematical and Computational Physics Mathematik Fluss Mathematik (DE-588)4489499-5 gnd Mittlere Krümmung (DE-588)4235959-4 gnd Regularität (DE-588)4049074-9 gnd |
| subject_GND | (DE-588)4489499-5 (DE-588)4235959-4 (DE-588)4049074-9 |
| title | Regularity Theory for Mean Curvature Flow |
| title_auth | Regularity Theory for Mean Curvature Flow |
| title_exact_search | Regularity Theory for Mean Curvature Flow |
| title_full | Regularity Theory for Mean Curvature Flow by Klaus Ecker |
| title_fullStr | Regularity Theory for Mean Curvature Flow by Klaus Ecker |
| title_full_unstemmed | Regularity Theory for Mean Curvature Flow by Klaus Ecker |
| title_short | Regularity Theory for Mean Curvature Flow |
| title_sort | regularity theory for mean curvature flow |
| topic | Mathematics Differential equations, partial Global differential geometry Differential Geometry Measure and Integration Partial Differential Equations Theoretical, Mathematical and Computational Physics Mathematik Fluss Mathematik (DE-588)4489499-5 gnd Mittlere Krümmung (DE-588)4235959-4 gnd Regularität (DE-588)4049074-9 gnd |
| topic_facet | Mathematics Differential equations, partial Global differential geometry Differential Geometry Measure and Integration Partial Differential Equations Theoretical, Mathematical and Computational Physics Mathematik Fluss Mathematik Mittlere Krümmung Regularität |
| url | https://doi.org/10.1007/978-0-8176-8210-1 |
| volume_link | (DE-604)BV036582883 |
| work_keys_str_mv | AT eckerklaus regularitytheoryformeancurvatureflow |