Minimal surfaces:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Heidelberg u. a.
Springer
2010
|
| Ausgabe: | Rev. and enl. 2. ed. |
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
339 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020774917&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XV, 688 S. Ill., graph. Darst. |
| ISBN: | 9783642116988 9783642116971 |
Internformat
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| 100 | 1 | |a Dierkes, Ulrich |d 1956- |e Verfasser |0 (DE-588)142533912 |4 aut | |
| 245 | 1 | 0 | |a Minimal surfaces |c Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny |
| 250 | |a Rev. and enl. 2. ed. | ||
| 264 | 1 | |a Heidelberg u. a. |b Springer |c 2010 | |
| 300 | |a XV, 688 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 339 | |
| 650 | 0 | 7 | |a Plateau-Problem |0 (DE-588)4174854-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Minimalfläche |0 (DE-588)4127814-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Minimalfläche |0 (DE-588)4127814-8 |D s |
| 689 | 0 | 1 | |a Plateau-Problem |0 (DE-588)4174854-2 |D s |
| 689 | 0 | 2 | |a Randwertproblem |0 (DE-588)4048395-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Hildebrandt, Stefan |d 1936-2015 |e Verfasser |0 (DE-588)119219050 |4 aut | |
| 700 | 1 | |a Sauvigny, Friedrich |d 1953- |e Verfasser |0 (DE-588)110083334 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-020774917 | |
Datensatz im Suchindex
| _version_ | 1819260188281012224 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
PART I. INTRODUCTION TO THE GEOMETRY OF SURFACES AND TO MINIMAL SURFACES
CHAPTER 1. DIFFERENTIAL GEOMETRY OF SURFACES IN THREE-DIMENSIONAL
EUCLIDEAN SPACE 3
1.1 SURFACES IN EUCLIDEAN SPACE 4
1.2 GAUSS MAP, WEINGARTEN MAP. FIRST, SECOND AND THIRD FUNDAMENTAL FORM.
MEAN CURVATURE AND GAUSS CURVATURE .. 9 1.3 GAUSS S REPRESENTATION
FORMULA, CHRISTOFFEL SYMBOLS, GAUSS-CODAZZI EQUATIONS, THEOREMA
EGREGIUM, MINDING S
FORMULA FOR THE GEODESIC CURVATURE 24
1.4 CONFORMAI PARAMETERS, GAUSS-BONNET THEOREM 33
1.5 COVARIANT DIFFERENTIATION. THE BELTRAMI OPERATOR 39
1.6 SCHOLIA 47
CHAPTER 2. MINIMAL SURFACES 53
2.1 FIRST VARIATION OF AREA. MINIMAL SURFACES 54
2.2 NONPARAMETRIC MINIMAL SURFACES 58
2.3 CONFORMAI REPRESENTATION AND ANALYTICITY OF NONPARAMETRIC MINIMAL
SURFACES 62
2.4 BERNSTEIN S THEOREM 66
2.5 TWO CHARACTERIZATIONS OF MINIMAL SURFACES 72
2.6 PARAMETRIC SURFACES IN CONFORMAI PARAMETERS. CONFORMAI
REPRESENTATION OF MINIMAL SURFACES. GENERAL DEFINITION OF MINIMAL
SURFACES 75
2.7 A FORMULA FOR THE MEAN CURVATURE 78
2.8 ABSOLUTE AND RELATIVE MINIMA OF AREA 82
2.9 SCHOLIA 86
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1007428619
DIGITALISIERT DURCH
IMAGE 2
VIII CONTENTS
CHAPTER 3. REPRESENTATION FORMULAS AND EXAMPLES OF MINIMAL SURFACES. 91
3.1 THE ADJOINT SURFACE. MINIMAL SURFACES AS ISOTROPIE CURVES IN C 3 .
ASSOCIATE MINIMAL SURFACES 93
3.2 BEHAVIOR OF MINIMAL SURFACES NEAR BRANCH POINTS 104 3.3
REPRESENTATION FORMULAS FOR MINIMAL SURFACES . . . I LL 3.4 BJOERLING S
PROBLEM. STRAIGHT LINES AND PLANAR LINES OF CURVATURE ON MINIMAL
SURFACES. SCHWARZIAN CHAINS 124
3.5 EXAMPLES OF MINIMAL SURFACES 141
3.5.1 CATENOID AND HELICOID 141
3.5.2 SCHERK S SECOND SURFACE: THE GENERAL MINIMAL SURFACE OF HELICOIDAL
TYPE 146
3.5.3 THE ENNEPER SURFACE 151
3.5.4 BOUR SURFACES 155
3.5.5 THOMSEN SURFACES 156
3.5.6 SCHERK S FIRST SURFACE 156
3.5.7 THE HENNEBERG SURFACE 166
3.5.8 CATALAN S SURFACE 171
3.5.9 SCHWARZ S SURFACE 182
3.6 COMPLETE MINIMAL SURFACES 183
3.7 OMISSIONS OF THE GAUSS MAP OF COMPLETE MINIMAL SURFACES . 190 3.8
SCHOLIA 200
COLOR PLATES 229
PART II. PLATEAU S PROBLEM
CHAPTER 4. THE PLATEAU PROBLEM AND THE PARTIALLY FREE BOUNDARY PROBLEM
239
4.1 AREA FUNCTIONAL VERSUS DIRICHLET INTEGRAL 246
4.2 RIGOROUS FORMULATION OF PLATEAU S PROBLEM AND OF THE MINIMIZATION
PROCESS 251
4.3 EXISTENCE PROOF, PART I: SOLUTION OF THE VARIATIONAL PROBLEM. 255
4.4 THE COURANT-LEBESGUE LEMMA 260
4.5 EXISTENCE PROOF, PART II: CONFORMALITY OF MINIMIZERS OF THE
DIRICHLET INTEGRAL 263
4.6 VARIANT OF THE EXISTENCE PROOF. THE PARTIALLY FREE BOUNDARY PROBLEM
275
4.7 BOUNDARY BEHAVIOR OF MINIMAL SURFACES WITH RECTIFIABLE BOUNDARIES
282
4.8 REFLECTION PRINCIPLES 289
4.9 UNIQUENESS AND NONUNIQUENESS QUESTIONS 292
4.10 ANOTHER SOLUTION OF PLATEAU S PROBLEM BY MINIMIZING AREA . 299 4.11
THE MAPPING THEOREMS OF RIEMANN AND LICHTENSTEIN 305 4.12 SOLUTION OF
PLATEAU S PROBLEM FOR NONRECTIFIABLE BOUNDARIES . 314
IMAGE 3
CONTENTS IX
4.13 PLATEAU S PROBLEM FOR CARIAN FUNCTIONALS 320
4.14 ISOPERIMETRIC INEQUALITIES 327
4.15 SCHOLIA 335
CHAPTER 5. STABLE MINIMAL- AND II-SURFACES 365
5.1 IF-SURFACES AND THEIR NORMALS 367
5.2 BONNET S MAPPING AND BONNET S SURFACE 371
5.3 THE SECOND VARIATION OF F FOR IF-SURFACES AND THEIR STABILITY 376
5.4 ON -STABLE IMMERSIONS OF CONSTANT MEAN CURVATURE 382 5.5 CURVATURE
ESTIMATES FOR STABLE AND IMMERSED CMC-SURFACES . 389 5.6 NITSCHE S
UNIQUENESS THEOREM AND FIELD-IMMERSIONS 395 5.7 SOME FINITENESS RESULTS
FOR PLATEAU S PROBLEM 407
5.8 SCHOLIA 420
CHAPTER 6. UNSTABLE MINIMAL SURFACES 425
6.1 COURANT S FUNCTION 0 426
6.2 COURANT S MOUNTAIN PASS LEMMA 438
6.3 UNSTABLE MINIMAL SURFACES IN A POLYGON 442
6.4 THE DOUGLAS FUNCTIONAL. CONVERGENCE THEOREMS FOR HARMONIC MAPPINGS
450
6.5 WHEN IS THE LIMES SUPERIOR OF A SEQUENCE OF PATHS AGAIN A PATH? 461
6.6 UNSTABLE MINIMAL SURFACES IN RECTIFIABLE BOUNDARIES 463 6.7 SCHOLIA
472
6.7.1 HISTORICAL REMARKS AND REFERENCES TO THE LITERATURE.. 472 6.7.2
THE THEOREM OF THE WALL FOR MINIMAL SURFACES IN TEXTBOOKS 473
6.7.3 SOURCES FOR THIS CHAPTER 474
6.7.4 MULTIPLY CONNECTED UNSTABLE MINIMAL SURFACES 474 6.7.5
QUASI-MINIMAL SURFACES 474
CHAPTER 7. GRAPHS WITH PRESCRIBED MEAN CURVATURE 493
7.1 //-SURFACES WITH A ONE-TO-ONE PROJECTION ONTO A PLANE, AND THE
NONPARAMETRIC DIRICHLET PROBLEM 494
7.2 UNIQUE SOLVABILITY OF PLATEAU S PROBLEM FOR CONTOURS WITH A
NONCONVEX PROJECTION ONTO A PLANE 508
7.3 MISCELLANEOUS ESTIMATES FOR NONPARAMETRIC II-SURFACES 516 7.4
SCHOLIA 529
CHAPTER 8. INTRODUCTION TO THE DOUGLAS PROBLEM 531
8.1 THE DOUGLAS PROBLEM. EXAMPLES AND MAIN RESULT 532
8.2 CONFORMALITY OF MINIMIZERS OF D IN {R) 538
IMAGE 4
X CONTENTS
8.3 COHESIVE SEQUENCES OF MAPPINGS 552
8.4 SOLUTION OF THE DOUGLAS PROBLEM 561
8.5 USEFUL MODIFICATIONS OF SURFACES 563
8.6 DOUGLAS CONDITION AND DOUGLAS PROBLEM 568
8.7 FURTHER DISCUSSION OF THE DOUGLAS CONDITION 578
8.8 EXAMPLES 581
8.9 SCHOLIA 584
PROBLEMS 587
APPENDIX 1. ON RELATIVE MINIMIZERS OF AREA AND ENERGY 589
APPENDIX 2. MINIMAL SURFACES IN HEISENBERG GROUPS 597
BIBLIOGRAPHY 599
INDEX 681
|
| any_adam_object | 1 |
| author | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Sauvigny, Friedrich 1953- |
| author_GND | (DE-588)142533912 (DE-588)119219050 (DE-588)110083334 |
| author_facet | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Sauvigny, Friedrich 1953- |
| author_role | aut aut aut |
| author_sort | Dierkes, Ulrich 1956- |
| author_variant | u d ud s h sh f s fs |
| building | Verbundindex |
| bvnumber | BV036859110 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)670417658 (DE-599)BVBBV036859110 |
| 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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| id | DE-604.BV036859110 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T14:43:26Z |
| institution | BVB |
| isbn | 9783642116988 9783642116971 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020774917 |
| oclc_num | 670417658 |
| open_access_boolean | |
| owner | DE-384 |
| owner_facet | DE-384 |
| physical | XV, 688 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spellingShingle | Dierkes, Ulrich 1956- Hildebrandt, Stefan 1936-2015 Sauvigny, Friedrich 1953- Minimal surfaces Grundlehren der mathematischen Wissenschaften Plateau-Problem (DE-588)4174854-2 gnd Randwertproblem (DE-588)4048395-2 gnd Minimalfläche (DE-588)4127814-8 gnd |
| subject_GND | (DE-588)4174854-2 (DE-588)4048395-2 (DE-588)4127814-8 |
| title | Minimal surfaces |
| title_auth | Minimal surfaces |
| title_exact_search | Minimal surfaces |
| title_full | Minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny |
| title_fullStr | Minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny |
| title_full_unstemmed | Minimal surfaces Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny |
| title_short | Minimal surfaces |
| title_sort | minimal surfaces |
| topic | Plateau-Problem (DE-588)4174854-2 gnd Randwertproblem (DE-588)4048395-2 gnd Minimalfläche (DE-588)4127814-8 gnd |
| topic_facet | Plateau-Problem Randwertproblem Minimalfläche |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020774917&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT dierkesulrich minimalsurfaces AT hildebrandtstefan minimalsurfaces AT sauvignyfriedrich minimalsurfaces |