Geodesic
Regular homotopy and total curvature II: sphere immersions into 3–space
Ekholm, Tobias1
1Department of mathematics, USC, Los Angeles CA 90803, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 493-512
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We consider properties of the total curvature functional on the space of 2–sphere immersions into 3–space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8π and we establish a non-injectivity result for local minima.

DOI : 10.2140/agt.2006.6.493
Keywords: immersion, regular homotopy, relatively isotopy tight, sphere eversion, total curvature

Ekholm, Tobias  1

1 Department of mathematics, USC, Los Angeles CA 90803, USA 
@article{10_2140_agt_2006_6_493,
     author = {Ekholm, Tobias},
     title = {Regular homotopy and total curvature {II:} sphere immersions into 3{\textendash}space},
     journal = {Algebraic and Geometric Topology},
     pages = {493--512},
     year = {2006},
     volume = {6},
     number = {1},
     doi = {10.2140/agt.2006.6.493},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.493/}
}
TY  - JOUR
AU  - Ekholm, Tobias
TI  - Regular homotopy and total curvature II: sphere immersions into 3–space
JO  - Algebraic and Geometric Topology
PY  - 2006
SP  - 493
EP  - 512
VL  - 6
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.493/
DO  - 10.2140/agt.2006.6.493
ID  - 10_2140_agt_2006_6_493
ER  - 
%0 Journal Article
%A Ekholm, Tobias
%T Regular homotopy and total curvature II: sphere immersions into 3–space
%J Algebraic and Geometric Topology
%D 2006
%P 493-512
%V 6
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.493/
%R 10.2140/agt.2006.6.493
%F 10_2140_agt_2006_6_493
Ekholm, Tobias. Regular homotopy and total curvature II: sphere immersions into 3–space. Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 493-512. doi: 10.2140/agt.2006.6.493

[1] T E Cecil, P J Ryan, Tight and taut immersions of manifolds, Research Notes in Mathematics 107, Pitman (Advanced Publishing Program) (1985)

[2] T Ekholm, Regular homotopy and total curvature I: circle immersions into surfaces, Algebr. Geom. Topol. 6 (2006) 459

[3] M Gromov, Partial differential relations, Ergebnisse series 9, Springer (1986)

[4] M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242

[5] N H Kuiper, Convex immersions of closed surfaces in $E^{3}$. Nonorientable closed surfaces in $E^{3}$ with minimal total absolute Gauss-curvature, Comment. Math. Helv. 35 (1961) 85

[6] N H Kuiper, W Meeks Iii, Total curvature for knotted surfaces, Invent. Math. 77 (1984) 25

[7] P Li, S T Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982) 269

[8] N Max, T Banchoff, Every sphere eversion has a quadruple point, from: "Contributions to analysis and geometry (Baltimore, Md., 1980)", Johns Hopkins Univ. Press (1981) 191

[9] S Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958) 281

[10] S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. $(2)$ 69 (1959) 327

Cité par Sources :