Geometry & Topology Volume 21, issue 4 (2017)

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Existence of minimizing Willmore Klein bottles in Euclidean four-space

Patrick Breuning, Jonas Hirsch and Elena Mäder-Baumdicker

Geometry & Topology 21 (2017) 2485–2526

DOI: 10.2140/gt.2017.21.2485

Bibliography

1	L V Ahlfors, Complex analysis: an introduction of the theory of analytic functions of one complex variable, McGraw-Hill (1966) MR0188405
2	M F Atiyah, N J Hitchin, I M Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978) 425 MR506229
3	T F Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974) 407 MR0377897
4	M Bauer, E Kuwert, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 2003 (2003) 553 MR1941840
5	L P Eisenhart, Minimal surfaces in Euclidean four-space, Amer. J. Math. 34 (1912) 215 MR1506152
6	J Eschenburg, R Tribuzy, Branch points of conformal mappings of surfaces, Math. Ann. 279 (1988) 621 MR926424
7	T Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984) 257 MR777909
8	J Hirsch, E Mäder-Baumdicker, A note on Willmore minimizing Klein bottles in Euclidean space, preprint (2016) arXiv:1606.04745
9	M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 MR0119214
10	M Koecher, A Krieg, Elliptische Funktionen und Modulformen, Springer (2007) MR2731246
11	R Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc. 17 (1987) 291 MR903735
12	R Kusner, Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral, from: "Elliptic and parabolic methods in geometry" (editors B Chow, R Gulliver, S Levy, J Sullivan), A K Peters (1996) 67 MR1417949
13	R Kusner, N Schmitt, The spinor representations of surfaces in space, preprint (1996) arXiv:dg-ga/9610005v1
14	E Kuwert, Y Li, W2,2–conformal immersions of a closed Riemann surface into ℝn, Comm. Anal. Geom. 20 (2012) 313 MR2928715
15	E Kuwert, R Schätzle, Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom. 93 (2013) 471 MR3024303
16	H Lapointe, Spectral properties of bipolar minimal surfaces in 𝕊4, Differential Geom. Appl. 26 (2008) 9 MR2393969
17	R Lashof, S Smale, On the immersion of manifolds in euclidean space, Ann. of Math. 68 (1958) 562 MR0103478
18	H B Lawson Jr., Complete minimal surfaces in S3, Ann. of Math. 92 (1970) 335 MR0270280
19	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 MR674407
20	F C Marques, A Neves, The Willmore conjecture, Jahresber. Dtsch. Math.-Ver. 116 (2014) 201 MR3280571
21	W S Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969) 143 MR0250331
22	J W Milnor, J D Stasheff, Characteristic classes, 76, Princeton Univ. Press (1974) MR0440554
23	R Miranda, Algebraic curves and Riemann surfaces, 5, Amer. Math. Soc. (1995) MR1326604
24	T Rivière, Analysis aspects of Willmore surfaces, Invent. Math. 174 (2008) 1 MR2430975
25	T Rivière, Variational principles for immersed surfaces with L2–bounded second fundamental form, J. Reine Angew. Math. 695 (2014) 41 MR3276154
26	S Salamon, Topics in four-dimensional Riemannian geometry, from: "Geometry seminar ‘Luigi Bianchi’" (editor E Vesentini), Lecture Notes in Math. 1022, Springer (1983) 33 MR728393
27	L Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993) 281 MR1243525
28	H Whitney, The self-intersections of a smooth n–manifold in 2n–space, Ann. of Math. 45 (1944) 220 MR0010274
29	P Wintgen, Sur l’inégalité de Chen–Willmore, C. R. Acad. Sci. Paris Sér. A–B 288 (1979) MR540375