Minimal submanifolds and related topics /:
The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribut...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore ; London :
World Scientific,
©2003.
|
| Schriftenreihe: | Nankai tracts in mathematics ;
v. 8. |
| Schlagworte: | |
| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribution of the Gauss image of a minimal surface in Euclidean 3-space, Simons' work for minimal graphic hypersurfaces, and author's own contributions to Bernstein type theorems for higher codimension. The author also introduces some related topics, such as submanifolds with parallel mean curvature, Weierstrass type representation for surfaces of mean curvature 1 in hyperbolic 3-space, and special Lagrangian submanifolds. |
| Beschreibung: | 1 online resource (viii, 262 pages) : illustrations |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9812564381 9789812564382 9789812386878 9812386874 |
Internformat
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| 245 | 1 | 0 | |a Minimal submanifolds and related topics / |c Yuanlong Xin. |
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| 505 | 0 | |a Ch. I. Introduction. 1.1. The second fundamental form. 1.2. The first variational formula. 1.3. Minimal submanifolds in euclidean space. 1.4. Minimal submanifolds in the sphere. 1.5. Examples. 1.6. Rigidity theorems. 1.7. exercises -- ch. II. Bernstein's theorem and its generalizations. 2.1. Gauss map. 2.2. The Weierstrass representation. 2.3. The value distribution of the image under the Gauss map. 2.4. Exercises -- ch. III. Weierstrass type representations. 3.1. The representation for surfaces of prescribed mean curvature. 3.2. The representation for CMC-1 surfaces in [symbol]. 3.3. Exercises -- ch. IV. Plateau's problem and Douglas-Rado solution. 4.1. Mathematical formulation. 4.2. The Dirichlet principle. 4.3. Proof of the main theorem -- ch. V. Minimal submanifolds of higher codimension. 5.1. Kähler geometry and Wirtinger's inequality. 5.2. Special Lagrangian submanifolds. 5.3. Exercises -- ch. VI. Stable minimal hypersurfaces. 6.1. The second variational formula. 6.2. Stable minimal hypersurfaces and applications. 6.3. A classification of certain stable minimal surfaces. 6.4. Stable cone and Bernstein's problem. 6.5. Curvature estimates for minimal hypersurfaces. 6.6. Exercises -- ch. VII. Bernstein type theorems for higher codimension. 7.1. Geometry of Grassmannian manifolds. 7.2. Harmonic Gauss maps. 7.3. Bernstein type theorems -- ch. VIII. Entire space-like submanifolds. 8.1. A Bochner type formula. 8.2. The Gauss image. 8.3. Estimates of the second fundamental form. 8.4. Completeness. 8.5. Bernstein's problem. 8.6. Final remarks. | |
| 520 | |a The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribution of the Gauss image of a minimal surface in Euclidean 3-space, Simons' work for minimal graphic hypersurfaces, and author's own contributions to Bernstein type theorems for higher codimension. The author also introduces some related topics, such as submanifolds with parallel mean curvature, Weierstrass type representation for surfaces of mean curvature 1 in hyperbolic 3-space, and special Lagrangian submanifolds. | ||
| 650 | 0 | |a Minimal submanifolds. |0 http://id.loc.gov/authorities/subjects/sh85129486 | |
| 650 | 0 | |a Geometry, Riemannian. |0 http://id.loc.gov/authorities/subjects/sh85054159 | |
| 650 | 6 | |a Sous-variétés minimales. | |
| 650 | 6 | |a Géométrie de Riemann. | |
| 650 | 7 | |a MATHEMATICS |x Geometry |x Differential. |2 bisacsh | |
| 650 | 0 | 7 | |a Minimal surfaces. |2 cct |
| 650 | 0 | 7 | |a Minimal submanifolds. |2 cct |
| 650 | 7 | |a Geometry, Riemannian |2 fast | |
| 650 | 7 | |a Minimal submanifolds |2 fast | |
| 776 | 0 | 8 | |i Print version: |a Xin, Y.L., 1943- |t Minimal submanifolds and related topics. |d Singapore ; River Edge, NJ : World Scientific, ©2003 |z 9812386874 |w (OCoLC)54701072 |
| 830 | 0 | |a Nankai tracts in mathematics ; |v v. 8. |0 http://id.loc.gov/authorities/names/n2001000055 | |
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Datensatz im Suchindex
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn228115549 |
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| _version_ | 1829094622602723328 |
| adam_text | |
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| author | Xin, Y. L., 1943- |
| author_GND | http://id.loc.gov/authorities/names/nr94022344 |
| author_facet | Xin, Y. L., 1943- |
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| author_sort | Xin, Y. L., 1943- |
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| contents | Ch. I. Introduction. 1.1. The second fundamental form. 1.2. The first variational formula. 1.3. Minimal submanifolds in euclidean space. 1.4. Minimal submanifolds in the sphere. 1.5. Examples. 1.6. Rigidity theorems. 1.7. exercises -- ch. II. Bernstein's theorem and its generalizations. 2.1. Gauss map. 2.2. The Weierstrass representation. 2.3. The value distribution of the image under the Gauss map. 2.4. Exercises -- ch. III. Weierstrass type representations. 3.1. The representation for surfaces of prescribed mean curvature. 3.2. The representation for CMC-1 surfaces in [symbol]. 3.3. Exercises -- ch. IV. Plateau's problem and Douglas-Rado solution. 4.1. Mathematical formulation. 4.2. The Dirichlet principle. 4.3. Proof of the main theorem -- ch. V. Minimal submanifolds of higher codimension. 5.1. Kähler geometry and Wirtinger's inequality. 5.2. Special Lagrangian submanifolds. 5.3. Exercises -- ch. VI. Stable minimal hypersurfaces. 6.1. The second variational formula. 6.2. Stable minimal hypersurfaces and applications. 6.3. A classification of certain stable minimal surfaces. 6.4. Stable cone and Bernstein's problem. 6.5. Curvature estimates for minimal hypersurfaces. 6.6. Exercises -- ch. VII. Bernstein type theorems for higher codimension. 7.1. Geometry of Grassmannian manifolds. 7.2. Harmonic Gauss maps. 7.3. Bernstein type theorems -- ch. VIII. Entire space-like submanifolds. 8.1. A Bochner type formula. 8.2. The Gauss image. 8.3. Estimates of the second fundamental form. 8.4. Completeness. 8.5. Bernstein's problem. 8.6. Final remarks. |
| ctrlnum | (OCoLC)228115549 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn228115549 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:36:03Z |
| institution | BVB |
| isbn | 9812564381 9789812564382 9789812386878 9812386874 |
| language | English |
| oclc_num | 228115549 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (viii, 262 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | World Scientific, |
| record_format | marc |
| series | Nankai tracts in mathematics ; |
| series2 | Nankai tracts in mathematics ; |
| spelling | Xin, Y. L., 1943- https://id.oclc.org/worldcat/entity/E39PCjGThfBWVwRXXFfwvFBWrC http://id.loc.gov/authorities/names/nr94022344 Minimal submanifolds and related topics / Yuanlong Xin. Singapore ; London : World Scientific, ©2003. 1 online resource (viii, 262 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier data file Nankai tracts in mathematics ; v. 8 Includes bibliographical references and index. Print version record. Ch. I. Introduction. 1.1. The second fundamental form. 1.2. The first variational formula. 1.3. Minimal submanifolds in euclidean space. 1.4. Minimal submanifolds in the sphere. 1.5. Examples. 1.6. Rigidity theorems. 1.7. exercises -- ch. II. Bernstein's theorem and its generalizations. 2.1. Gauss map. 2.2. The Weierstrass representation. 2.3. The value distribution of the image under the Gauss map. 2.4. Exercises -- ch. III. Weierstrass type representations. 3.1. The representation for surfaces of prescribed mean curvature. 3.2. The representation for CMC-1 surfaces in [symbol]. 3.3. Exercises -- ch. IV. Plateau's problem and Douglas-Rado solution. 4.1. Mathematical formulation. 4.2. The Dirichlet principle. 4.3. Proof of the main theorem -- ch. V. Minimal submanifolds of higher codimension. 5.1. Kähler geometry and Wirtinger's inequality. 5.2. Special Lagrangian submanifolds. 5.3. Exercises -- ch. VI. Stable minimal hypersurfaces. 6.1. The second variational formula. 6.2. Stable minimal hypersurfaces and applications. 6.3. A classification of certain stable minimal surfaces. 6.4. Stable cone and Bernstein's problem. 6.5. Curvature estimates for minimal hypersurfaces. 6.6. Exercises -- ch. VII. Bernstein type theorems for higher codimension. 7.1. Geometry of Grassmannian manifolds. 7.2. Harmonic Gauss maps. 7.3. Bernstein type theorems -- ch. VIII. Entire space-like submanifolds. 8.1. A Bochner type formula. 8.2. The Gauss image. 8.3. Estimates of the second fundamental form. 8.4. Completeness. 8.5. Bernstein's problem. 8.6. Final remarks. The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribution of the Gauss image of a minimal surface in Euclidean 3-space, Simons' work for minimal graphic hypersurfaces, and author's own contributions to Bernstein type theorems for higher codimension. The author also introduces some related topics, such as submanifolds with parallel mean curvature, Weierstrass type representation for surfaces of mean curvature 1 in hyperbolic 3-space, and special Lagrangian submanifolds. Minimal submanifolds. http://id.loc.gov/authorities/subjects/sh85129486 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Sous-variétés minimales. Géométrie de Riemann. MATHEMATICS Geometry Differential. bisacsh Minimal surfaces. cct Minimal submanifolds. cct Geometry, Riemannian fast Minimal submanifolds fast Print version: Xin, Y.L., 1943- Minimal submanifolds and related topics. Singapore ; River Edge, NJ : World Scientific, ©2003 9812386874 (OCoLC)54701072 Nankai tracts in mathematics ; v. 8. http://id.loc.gov/authorities/names/n2001000055 |
| spellingShingle | Xin, Y. L., 1943- Minimal submanifolds and related topics / Nankai tracts in mathematics ; Ch. I. Introduction. 1.1. The second fundamental form. 1.2. The first variational formula. 1.3. Minimal submanifolds in euclidean space. 1.4. Minimal submanifolds in the sphere. 1.5. Examples. 1.6. Rigidity theorems. 1.7. exercises -- ch. II. Bernstein's theorem and its generalizations. 2.1. Gauss map. 2.2. The Weierstrass representation. 2.3. The value distribution of the image under the Gauss map. 2.4. Exercises -- ch. III. Weierstrass type representations. 3.1. The representation for surfaces of prescribed mean curvature. 3.2. The representation for CMC-1 surfaces in [symbol]. 3.3. Exercises -- ch. IV. Plateau's problem and Douglas-Rado solution. 4.1. Mathematical formulation. 4.2. The Dirichlet principle. 4.3. Proof of the main theorem -- ch. V. Minimal submanifolds of higher codimension. 5.1. Kähler geometry and Wirtinger's inequality. 5.2. Special Lagrangian submanifolds. 5.3. Exercises -- ch. VI. Stable minimal hypersurfaces. 6.1. The second variational formula. 6.2. Stable minimal hypersurfaces and applications. 6.3. A classification of certain stable minimal surfaces. 6.4. Stable cone and Bernstein's problem. 6.5. Curvature estimates for minimal hypersurfaces. 6.6. Exercises -- ch. VII. Bernstein type theorems for higher codimension. 7.1. Geometry of Grassmannian manifolds. 7.2. Harmonic Gauss maps. 7.3. Bernstein type theorems -- ch. VIII. Entire space-like submanifolds. 8.1. A Bochner type formula. 8.2. The Gauss image. 8.3. Estimates of the second fundamental form. 8.4. Completeness. 8.5. Bernstein's problem. 8.6. Final remarks. Minimal submanifolds. http://id.loc.gov/authorities/subjects/sh85129486 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Sous-variétés minimales. Géométrie de Riemann. MATHEMATICS Geometry Differential. bisacsh Minimal surfaces. cct Minimal submanifolds. cct Geometry, Riemannian fast Minimal submanifolds fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85129486 http://id.loc.gov/authorities/subjects/sh85054159 |
| title | Minimal submanifolds and related topics / |
| title_auth | Minimal submanifolds and related topics / |
| title_exact_search | Minimal submanifolds and related topics / |
| title_full | Minimal submanifolds and related topics / Yuanlong Xin. |
| title_fullStr | Minimal submanifolds and related topics / Yuanlong Xin. |
| title_full_unstemmed | Minimal submanifolds and related topics / Yuanlong Xin. |
| title_short | Minimal submanifolds and related topics / |
| title_sort | minimal submanifolds and related topics |
| topic | Minimal submanifolds. http://id.loc.gov/authorities/subjects/sh85129486 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Sous-variétés minimales. Géométrie de Riemann. MATHEMATICS Geometry Differential. bisacsh Minimal surfaces. cct Minimal submanifolds. cct Geometry, Riemannian fast Minimal submanifolds fast |
| topic_facet | Minimal submanifolds. Geometry, Riemannian. Sous-variétés minimales. Géométrie de Riemann. MATHEMATICS Geometry Differential. Minimal surfaces. Geometry, Riemannian Minimal submanifolds |
| work_keys_str_mv | AT xinyl minimalsubmanifoldsandrelatedtopics |