A course in minimal surfaces:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2011
|
| Schriftenreihe: | Graduate studies in mathematics
121 |
| Schlagworte: |
Differential geometry
> Classical differential geometry
> Minimal surfaces, surfaces with prescribed mean curvature
Differential geometry
> Global differential geometry
> Immersions (minimal, prescribed curvature, tight, etc.)
Global analysis, analysis on manifolds
> Variational problems in infinite-dimensional spaces
> Applications to minimal surfaces (problems in two independent variables)
|
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XII, 313 S. Ill., graph. Darst. |
| ISBN: | 9780821853238 |
Internformat
MARC
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| 100 | 1 | |a Colding, Tobias H. |e Verfasser |0 (DE-588)1029523185 |4 aut | |
| 245 | 1 | 0 | |a A course in minimal surfaces |c Tobias Holck Colding ; William P. Minicozzi II |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2011 | |
| 300 | |a XII, 313 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 121 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Minimal surfaces | |
| 650 | 7 | |a Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces |2 msc | |
| 650 | 7 | |a Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature |2 msc | |
| 650 | 7 | |a Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.) |2 msc | |
| 650 | 7 | |a Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables) |2 msc | |
| 650 | 7 | |a Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds |2 msc | |
| 650 | 7 | |a Manifolds and cell complexes -- Topological manifolds -- Topology of general 3-manifolds |2 msc | |
| 650 | 7 | |a Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations |2 msc | |
| 650 | 7 | |a Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations |2 msc | |
| 650 | 7 | |a Relativity and gravitational theory -- General relativity -- Black holes |2 msc | |
| 650 | 0 | 7 | |a Minimalfläche |0 (DE-588)4127814-8 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Graduate studies in mathematics |v 121 |w (DE-604)BV009739289 |9 121 | |
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Datensatz im Suchindex
| _version_ | 1804148021518139392 |
|---|---|
| adam_text | Contents
Preface
їх
Chapter
1.
The Beginning of the Theory
1
§1.
The Minimal Surface Equation and Minimal Submanifolds
1
§2.
Examples of Minimal Surfaces in
Ш3
8
§3.
Consequences of the First Variation Formula
18
§4.
The Gauss Map
28
§5.
The Theorem of Bernstein
29
§6.
The
Weierstrass
Representation
33
§7.
The Strong Maximum Principle
37
§8.
Second Variation Formula, Morse Index, and Stability
38
§9.
Multi-valued Graphs
50
§10.
Local Examples of Multi-valued Graphs
51
Appendix: The Harnack Inequality
60
Appendix: The Bochner formula
60
Chapter
2.
Curvature Estimates and Consequences
65
§1.
Simons Inequality
66
§2.
Small Energy Curvature Estimates for Minimal Surfaces
72
§3.
Curvature and Area
76
§4.
Lp Bounds of A 2 for Stable Hypersurfaces
85
§5.
Bernstein Theorems and Curvature Estimates
87
§6.
The General Minimal Graph Equation
88
§7.
Almost Stability
92
vi
Contents
§8.
Sublinear
Growth of the Separation
97
§9.
Minimal Cones
101
Chapter
3.
Weak Convergence, Compactness and Applications
105
§1.
The Theory of
Var
ifolds
106
§2.
The Sobolev Inequality
113
§3.
The Weak Bernstein-Type Theorem
119
§4.
General Constructions
120
§5.
Finite Dimensionality
124
§6.
Bubble Convergence Implies Varifold Convergence
128
Chapter
4.
Existence Results
133
81
The Plateau Problem
133
52
The Dirichlet Problem
137
§3.
The Solution to the Plateau Problem
142
§4.
Branch Points
147
§5.
Harmonic Maps
148
§6.
Existence of Minimal Spheres in a Homotopy Class
153
Chapter
5.
Min-max
Constructions
163
§1.
Sweepouts by Curves
163
§2.
Birkhoff s Curve Shortening Process
164
§3.
Existence of Closed Geodesies and the Width
171
§4.
Harmonic Replacement
176
§5.
Minimal Spheres and the Width
190
Chapter
6.
Embedded Solutions of the Plateau problem
201
§1.
Unique Continuation
201
§2.
Local Description of Nodal and Critical Sets
207
§3.
Absence of True Branch Points
215
§4.
Absence of False Branch Points
217
§5.
Embedded Solutions of the Plateau Problem
224
Chapter
7.
Minimal Surfaces in Three-Manifolds
233
§1.
The Minimal Surface Equation in a Three-Manifold
233
§2.
Hersch s and Yang and Yau s Theorems
239
§3.
The Reilly Formula
242
§4.
Choi and Wang s Lower Bound for
λχ
243
Contents
vii
§5.
Compactness Theorems with A Priori Bounds
244
§6.
The Positive Mass Theorem
251
§7.
Extinction of
Ricci
Flow
255
Chapter
8.
The Structure of Embedded Minimal Surfaces
261
§1.
Disks that are Double-spiral Staircases
261
§2.
One-sided Curvature Estimate
276
§3.
Generalized Nitsche Conjecture
278
§4.
Calabi-Yau Conjectures for Embedded Surfaces
282
§5.
Embedded Minimal Surfaces with Finite Genus
288
Exercises
295
Bibliography
299
Index
311
Minimal
surfaces
date back to
Euler
and
Lagrange
and the beginning of the calculus
of variations. Many of the techniques devel¬
oped have played key roles in geometry
and partial differential equations. Examples
include
monotonicity
and tangent cone anal¬
ysis originating in the regularity theory for
minimal surfaces, estimates for nonlinear
equations based on the maximum principle
----------------------
arising in Bernstein s classical work, and even Lebesgue s definition of the integral
that he developed in his thesis on the Plateau problem for minimal surfaces.
This book starts with the classical theory of minimal surfaces and ends up with
current research topics. Of the various ways of approaching minimal surfaces (from
complex analysis, PDE, or geometric measure theory), the authors have chosen
to focus on the PDE aspects of the theory. The book also contains some of the
applications of minimal surfaces to other fields including low dimensional topology,
general relativity, and materials science.
The only prerequisites needed for this book are a basic knowledge of Riemannian
geometry and some familiarity with the maximum principle.
|
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| isbn | 9780821853238 |
| language | English |
| lccn | 2010044373 |
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| series | Graduate studies in mathematics |
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| spelling | Colding, Tobias H. Verfasser (DE-588)1029523185 aut A course in minimal surfaces Tobias Holck Colding ; William P. Minicozzi II Providence, RI American Mathematical Society 2011 XII, 313 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 121 Includes bibliographical references and index Minimal surfaces Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces msc Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature msc Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.) msc Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables) msc Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds msc Manifolds and cell complexes -- Topological manifolds -- Topology of general 3-manifolds msc Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations msc Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations msc Relativity and gravitational theory -- General relativity -- Black holes msc Minimalfläche (DE-588)4127814-8 gnd rswk-swf Minimalfläche (DE-588)4127814-8 s DE-604 Minicozzi, William P. 1967- Verfasser (DE-588)1029523835 aut Graduate studies in mathematics 121 (DE-604)BV009739289 121 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024186225&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024186225&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Colding, Tobias H. Minicozzi, William P. 1967- A course in minimal surfaces Graduate studies in mathematics Minimal surfaces Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces msc Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature msc Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.) msc Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables) msc Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds msc Manifolds and cell complexes -- Topological manifolds -- Topology of general 3-manifolds msc Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations msc Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations msc Relativity and gravitational theory -- General relativity -- Black holes msc Minimalfläche (DE-588)4127814-8 gnd |
| subject_GND | (DE-588)4127814-8 |
| title | A course in minimal surfaces |
| title_auth | A course in minimal surfaces |
| title_exact_search | A course in minimal surfaces |
| title_full | A course in minimal surfaces Tobias Holck Colding ; William P. Minicozzi II |
| title_fullStr | A course in minimal surfaces Tobias Holck Colding ; William P. Minicozzi II |
| title_full_unstemmed | A course in minimal surfaces Tobias Holck Colding ; William P. Minicozzi II |
| title_short | A course in minimal surfaces |
| title_sort | a course in minimal surfaces |
| topic | Minimal surfaces Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces msc Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature msc Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.) msc Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables) msc Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds msc Manifolds and cell complexes -- Topological manifolds -- Topology of general 3-manifolds msc Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations msc Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations msc Relativity and gravitational theory -- General relativity -- Black holes msc Minimalfläche (DE-588)4127814-8 gnd |
| topic_facet | Minimal surfaces Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.) Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables) Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds Manifolds and cell complexes -- Topological manifolds -- Topology of general 3-manifolds Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations Relativity and gravitational theory -- General relativity -- Black holes Minimalfläche |
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