Global analysis of minimal surfaces:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u. a.
Springer
2010
|
| Edition: | Rev. and enl. 2. ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
341 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 6537 S. Ill., graph. Darst. |
| ISBN: | 9783642117060 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | IMAGE 1
CONTENTS
PART I. FREE BOUNDARIES AND BERNSTEIN THEOREMS
CHAPTER 1. MINIMAL SURFACES WITH SUPPORTING HALF-PLANES 3
1.1 AN EXPERIMENT 4
1.2 EXAMPLES OF MINIMAL SURFACES WITH CUSPS ON THE SUPPORTING SURFACE 7
1.3 SETUP OF THE PROBLEM. PROPERTIES OF STATIONARY SOLUTIONS.... 11 1.4
CLASSIFICATION OF THE CONTACT SETS 13
1.5 NONPARAMETRIC REPRESENTATION, UNIQUENESS, AND SYMMETRY OF SOLUTIONS
18
1.6 ASYMPTOTIC EXPANSIONS FOR SURFACES OF CUSP-TYPES I AND III. MINIMA
OF DIRICHLET S INTEGRAL 21
1.7 ASYMPTOTIC EXPANSIONS FOR SURFACES OF THE TONGUE/LOOP- TYPE II 23
1.8 FINAL RESULTS ON THE SHAPE OF THE TRACE. ABSENCE OF CUSPS. OPTIMAL
BOUNDARY REGULARITY 26
1.9 PROOF OF THE REPRESENTATION THEOREM 28
1.10 SCHOLIA 34
CHAPTER 2. EMBEDDED MINIMAL SURFACES WITH PARTIALLY FREE BOUNDARIES . 37
2.1 THE GEOMETRIC SETUP 38
2.2 INCLUSION AND MONOTONICITY OF THE FREE BOUNDARY VALUES . . .. 44 2.3
A MODIFICATION OF THE KNESER-RADO THEOREM 50
2.4 PROPERTIES OF THE GAUSS MAP, AND STABLE SURFACES 52
2.5 UNIQUENESS OF MINIMAL SURFACES THAT LIE ON ONE SIDE OF THE
SUPPORTING SURFACE 60
2.6 UNIQUENESS OF FREELY STABLE MINIMAL SURFACES 66
2.7 ASYMPTOTIC EXPANSIONS 74
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1007548401
DIGITALISIERT DURCH
IMAGE 2
VIII CONTENTS
2.8 EDGE CREEPING 86
2.9 EMBEDDED MINIMIZERS FOR NONSMOOTH SUPPORTING SURFACES .. 96 2.10 A
BERNSTEIN THEOREM FOR MINIMAL SURFACES IN A WEDGE 108 2.11 SCHOLIA 126
CHAPTER 3. BERNSTEIN THEOREMS AND RELATED RESULTS 135
3.1 ENTIRE AND EXTERIOR MINIMAL GRAPHS OF CONTROLLED GROWTH .. 137 3.1.1
JOERGENS S THEOREM 137
3.1.2 ASYMPTOTIC BEHAVIOUR FOR SOLUTIONS OF LINEAR AND QUASILINEAR
EQUATIONS, MOSER S BERNSTEIN THEOREM . .. 140 3.1.3 THE INTERIOR
GRADIENT ESTIMATE AND CONSEQUENCES . .. 144 3.2 FIRST AND SECOND
VARIATION FORMULAE 145
3.2.1 FIRST AND SECOND VARIATION OF THE AREA INTEGRAL 146 3.2.2 FIRST
AND SECOND VARIATION FORMULAE FOR SINGULAR MINIMAL SURFACES 152
3.3 SOME GEOMETRIC IDENTITIES 156
3.3.1 COVARIANT DERIVATIVES OF TENSOR FIELDS 159
3.3.2 SIMONS S IDENTITY AND JACOBI S FIELD EQUATION 161 3.4 NONEXISTENCE
OF STABLE CONES AND INTEGRAL CURVATURE ESTIMATES. FURTHER BERNSTEIN
THEOREMS 163
3.4.1 STABILITY OF MINIMAL CONES 164
3.4.2 NONEXISTENCE OF STABLE CONES 172
3.4.3 INTEGRAL CURVATURE ESTIMATES FOR MINIMAL AND A-MINIMAL
HYPERSURFACES. FURTHER BERNSTEIN THEOREMS 180
3.5 MONOTONICITY AND MEAN VALUE FORMULAE. MICHAEL-SIMON INEQUALITIES 198
3.6 POINTWISE CURVATURE ESTIMATES 217
3.7 SCHOLIA 236
3.7.1 REFERENCES TO THE LITERATURE ON BERNSTEIN S THEOREM AND CURVATURE
ESTIMATES FOR N = 2 236
3.7.2 BERNSTEIN THEOREMS AND CURVATURE ESTIMATES FOR N 3 DIMENSIONS
238
3.7.3 BERNSTEIN THEOREMS IN HIGHER CODIMENSIONS 242 3.7.4 SOBOLEV
INEQUALITIES 245
PART II. GLOBAL ANALYSIS OF MINIMAL SURFACES
CHAPTER 4. THE GENERAL PROBLEM OF PLATEAU: ANOTHER APPROACH 249
4.1 THE GENERAL PROBLEM OF PLATEAU. FORMULATION AND EXAMPLES 249 4.2 A
GEOMETRIC APPROACH TO TEICHMUELLER THEORY OF ORIENTED SURFACES 255
4.3 SYMMETRIC RIEMANN SURFACES AND THEIR TEICHMUELLER SPACES . 263
IMAGE 3
CONTENTS IX
4.4 THE MUMFORD COMPACTNESS THEOREM 271
4.5 THE VARIATIONAL PROBLEM 276
4.6 EXISTENCE RESULTS FOR THE GENERAL PROBLEM OF PLATEAU IN R 3 . 285
4.7 SCHOLIA 296
CHAPTER 5. THE INDEX THEOREMS FOR MINIMAL SURFACES OF ZERO AND HIGHER
GENUS 299
5.1 INTRODUCTION 299
5.2 THE STATEMENT OF THE INDEX THEOREM OF GENUS ZERO 302 5.3
STRATIFICATION OF HARMONIC SURFACES BY SINGULARITY TYPE . . .. 304 5.4
STRATIFICATION OF HARMONIC SURFACES WITH REGULAR BOUNDARIES BY
SINGULARITY TYPE 318
5.5 THE INDEX THEOREM FOR CLASSICAL MINIMAL SURFACES 324 5.6 THE FORCED
JACOBI FIELDS 329
5.7 SOME THEOREMS ON THE LINEAR ALGEBRA OF FREDHOLM MAPS . .. 341 5.8
GENERIC FINITENESS, STABILITY, AND THE STRATIFICATION OF THE SETS M O
347
5.9 THE INDEX THEOREM FOR HIGHER GENUS MINIMAL SURFACES STATEMENT AND
PRELIMINARIES 353
5.10 REVIEW OF SOME BASIC RESULTS IN RIEMANN SURFACE THEORY .. 354 5.11
VECTOR BUNDLES OVER TEICHMUELLER SPACE 359
5.12 SOME RESULTS ON MAXIMAL IDEALS IN SOBOLEV ALGEBRAS OF HOLOMORPHIC
FUNCTIONS 364
5.13 MINIMAL SURFACES AS ZEROS OF A VECTOR FIELD, AND THE CONFORMALITY
OPERATORS 365
5.14 THE CORANK OF THE PARTIAL CONFORMALITY OPERATORS 369 5.15 THE
CORANK OF THE COMPLETE CONFORMALITY OPERATORS 377 5.16 MANIFOLDS OF
HARMONIC SURFACES OF PRESCRIBED BRANCHING TYPE 380
5.17 THE PROOF OF THE INDEX THEOREM 385
5.18 SCHOLIA 399
CHAPTER 6. EULER CHARACTERISTIC AND MORSE THEORY FOR MINIMAL SURFACES
401
6.1 FREDHOLM VECTOR FIELDS 402
6.2 THE GRADIENT VECTOR FIELD ASSOCIATED TO PLATEAU S PROBLEM . 405 6.3
THE EULER CHARACTERISTIC X(W A ) OF W A 411
6.4 THE SARD-BROWN THEOREM FOR FUNCTIONALS 423
6.5 THE MORSE LEMMA 424
6.6 THE NORMAL FORM OF DIRICHLET S ENERGY ABOUT A GENERIC MINIMAL
SURFACE IN M 3 436
6.7 THE LOCAL WINDING NUMBER OF W A ABOUT A GENERICALLY BRANCHED MINIMAL
SURFACE IN E 3 442
6.8 SCHOLIA 447
IMAGE 4
X CONTENTS
6.8.1 HISTORICAL REMARKS AND REFERENCES TO THE LITERATURE.. 447 6.8.2 ON
THE GENERIC NONDEGENERACY OF CLOSED MINIMAL SURFACES IN RIEMANNIAN
MANIFOLDS AND MORSE THEORY 449
BIBLIOGRAPHY 477
INDEX 531
|
| any_adam_object | 1 |
| author | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Tromba, Anthony J. 1943- |
| author_GND | (DE-588)142533912 (DE-588)119219050 (DE-588)142684996 |
| author_facet | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Tromba, Anthony J. 1943- |
| author_role | aut aut aut |
| author_sort | Dierkes, Ulrich 1956- |
| author_variant | u d ud s h sh a j t aj ajt |
| building | Verbundindex |
| bvnumber | BV036859222 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)670417695 (DE-599)BVBBV036859222 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Rev. and enl. 2. ed. |
| format | Book |
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| indexdate | 2024-07-09T22:49:34Z |
| institution | BVB |
| isbn | 9783642117060 |
| language | English |
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| series | Grundlehren der mathematischen Wissenschaften |
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| spelling | Dierkes, Ulrich 1956- Verfasser (DE-588)142533912 aut Global analysis of minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba Rev. and enl. 2. ed. Berlin u. a. Springer 2010 XVI, 6537 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 341 Globale Analysis (DE-588)4021285-3 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Minimalfläche (DE-588)4127814-8 gnd rswk-swf Minimalfläche (DE-588)4127814-8 s Randwertproblem (DE-588)4048395-2 s Globale Analysis (DE-588)4021285-3 s DE-604 Hildebrandt, Stefan 1936-2015 Verfasser (DE-588)119219050 aut Tromba, Anthony J. 1943- Verfasser (DE-588)142684996 aut Grundlehren der mathematischen Wissenschaften 341 (DE-604)BV000000395 341 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020775024&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Tromba, Anthony J. 1943- Global analysis of minimal surfaces Grundlehren der mathematischen Wissenschaften Globale Analysis (DE-588)4021285-3 gnd Randwertproblem (DE-588)4048395-2 gnd Minimalfläche (DE-588)4127814-8 gnd |
| subject_GND | (DE-588)4021285-3 (DE-588)4048395-2 (DE-588)4127814-8 |
| title | Global analysis of minimal surfaces |
| title_auth | Global analysis of minimal surfaces |
| title_exact_search | Global analysis of minimal surfaces |
| title_full | Global analysis of minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba |
| title_fullStr | Global analysis of minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba |
| title_full_unstemmed | Global analysis of minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba |
| title_short | Global analysis of minimal surfaces |
| title_sort | global analysis of minimal surfaces |
| topic | Globale Analysis (DE-588)4021285-3 gnd Randwertproblem (DE-588)4048395-2 gnd Minimalfläche (DE-588)4127814-8 gnd |
| topic_facet | Globale Analysis Randwertproblem Minimalfläche |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020775024&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT dierkesulrich globalanalysisofminimalsurfaces AT hildebrandtstefan globalanalysisofminimalsurfaces AT trombaanthonyj globalanalysisofminimalsurfaces |