Variational methods in geometry. (English) Zbl 1371.49038
Author’s abstract: Variational principles are ubiquitous in nature. Many geometric objects such as geodesics or minimal surfaces allow variational characterizations. We recall some basic ideas in the calculus of variations, also relevant for some of the most advanced research in the field today, and show how a subtle variation of standard methods can lead to surprising improvements, with numerous applications.
Reviewer: Constantin Zălinescu (Iaşi)
MSC:
| 49Q05 | Minimal surfaces and optimization |
| 49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |
| 58E20 | Harmonic maps, etc. |
| 53C22 | Geodesics in global differential geometry |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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