Book Review
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MathSciNet review:
4076540
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Book Information:
Author:
John Douglas Moore
Title:
Introduction to Global Analysis. Minimal Surfaces in Riemannian Manifolds
Additional book information:
Graduate Studies in Mathematics, Vol. 187,
American Mathematical Society,
Providence, RI,
2017,
xiv+368 pp.,
ISBN 978-1-4704-2950-8,
$83.00
R. Abraham, Bumpy metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–3. MR 0271994
Tobias H. Colding and William P. Minicozzi II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537–2586. MR 2460871, DOI 10.2140/gt.2008.12.2537
Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
H. Blaine Lawson Jr., Lectures on minimal submanifolds. Vol. I, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish, Inc., Wilmington, Del., 1980. MR 576752
William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
John Douglas Moore, Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5193–5256. MR 2238914, DOI 10.1090/S0002-9947-06-04317-0
John Douglas Moore, Introduction to global analysis, Graduate Studies in Mathematics, vol. 187, American Mathematical Society, Providence, RI, 2017. Minimal surfaces in Riemannian manifolds. MR 3729450, DOI 10.1090/gsm/187
Marston Morse, The calculus of variations in the large, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR 1451874, DOI 10.1090/coll/018
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Joaquín Pérez, A new golden age of minimal surfaces, Notices Amer. Math. Soc. 64 (2017), no. 4, 347–358. MR 3617312, DOI 10.1090/noti1500
Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 626027
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI 10.1090/S0002-9947-1982-0654854-8
Richard M. Schoen, Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR 765241, DOI 10.1007/978-1-4612-1110-5_{1}7
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI 10.2307/1970949
Brian White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR 1101226, DOI 10.1512/iumj.1991.40.40008
References
- R. Abraham, Bumpy metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amererican Mathematical Society, Providence, R.I., 1970, pp. 1–3. MR 0271994
- Tobias H. Colding and William P. Minicozzi II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537–2586. MR 2460871, DOI 10.2140/gt.2008.12.2537
- Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
- H. Blaine Lawson Jr., Lectures on minimal submanifolds. Vol. I, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish, Inc., Wilmington, Del., 1980. MR 576752
- William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
- Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
- J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
- John Douglas Moore, Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5193–5256. MR 2238914, DOI 10.1090/S0002-9947-06-04317-0
- John Douglas Moore, Introduction to global analysis: Minimal surfaces in Riemannian manifolds, Graduate Studies in Mathematics, vol. 187, American Mathematical Society, Providence, RI, 2017. MR 3729450
- Marston Morse, The calculus of variations in the large, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR 1451874
- Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
- Joaquín Pérez, A new golden age of minimal surfaces, Notices Amer. Math. Soc. 64 (2017), no. 4, 347–358. MR 3617312, DOI 10.1090/noti1500
- Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 626027
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
- J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI 10.2307/1998902
- Richard M. Schoen, Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR 765241, DOI 10.1007/978-1-4612-1110-5_17
- R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
- S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
- Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI 10.2307/1970949
- Brian White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR 1101226, DOI 10.1512/iumj.1991.40.40008
Review Information:
Reviewer:
Tobias Holck Colding
Affiliation:
Mathematics Department, MIT, Cambridge, Massachusetts
Email:
colding@math.mit.edu
Journal:
Bull. Amer. Math. Soc.
57 (2020), 353-356
DOI:
https://doi.org/10.1090/bull/1689
Published electronically:
January 3, 2020
Review copyright:
© Copyright 2020
American Mathematical Society
