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Degree Theory of Immersed Hypersurfaces.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2020Copyright date: ©2020Edition: 1st edDescription: 1 online resource (74 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470461485
Subject(s): Genre/Form: Additional physical formats: Print version:: Degree Theory of Immersed HypersurfacesDDC classification:
  • 516.3/73
LOC classification:
  • QA671 .R674 2020
Online resources:
Contents:
Cover -- Title page -- Chapter 1. Introduction -- 1.1. General -- 1.2. Background -- 1.3. Applications -- Acknowledgments -- Chapter 2. Degree theory -- 2.1. The manifold of immersions and its tangent bundle -- 2.2. Curvature as a vector field -- 2.3. Simplicity -- 2.4. Surjectivity -- 2.5. Finite dimensional sections -- 2.6. Extensions -- 2.7. Orientation - the finite-dimensional case -- 2.8. Orientation - the infinite-dimensional case -- 2.9. Constructing the degree -- 2.10. Varying the metric -- Chapter 3. Applications -- 3.1. The generalised Simons' formula -- 3.2. Prescribed mean curvature -- 3.3. Calculating the Degree -- 3.4. Extrinstic Curvature -- 3.5. Special Lagrangian curvature -- 3.6. Extrinsic curvature in two dimensions -- Appendix A. Weakly smooth maps -- Appendix B. Prime immersions -- Bibliography -- Back Cover.
Summary: The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to -\chi(M), where \chi(M) is the Euler characteristic of the ambient manifold M.
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Cover -- Title page -- Chapter 1. Introduction -- 1.1. General -- 1.2. Background -- 1.3. Applications -- Acknowledgments -- Chapter 2. Degree theory -- 2.1. The manifold of immersions and its tangent bundle -- 2.2. Curvature as a vector field -- 2.3. Simplicity -- 2.4. Surjectivity -- 2.5. Finite dimensional sections -- 2.6. Extensions -- 2.7. Orientation - the finite-dimensional case -- 2.8. Orientation - the infinite-dimensional case -- 2.9. Constructing the degree -- 2.10. Varying the metric -- Chapter 3. Applications -- 3.1. The generalised Simons' formula -- 3.2. Prescribed mean curvature -- 3.3. Calculating the Degree -- 3.4. Extrinstic Curvature -- 3.5. Special Lagrangian curvature -- 3.6. Extrinsic curvature in two dimensions -- Appendix A. Weakly smooth maps -- Appendix B. Prime immersions -- Bibliography -- Back Cover.

The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to -\chi(M), where \chi(M) is the Euler characteristic of the ambient manifold M.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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