Geometry & Topology

Existence of minimizing Willmore Klein bottles in Euclidean four-space

Abstract

Let $K = ℝP2 ♯ ℝP2$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles $f : K → ℝn$ for $n ≥ 4$ is attained by a smooth embedded Klein bottle. We know from work of M W Hirsch and W S Massey that there are three distinct regular homotopy classes of immersions $f : K → ℝ4$, each one containing an embedding. One is characterized by the property that it contains the minimizer just mentioned. For the other two regular homotopy classes we show $W(f) ≥ 8π$. We give a classification of the minimizers of these two regular homotopy classes. In particular, we prove the existence of infinitely many distinct embedded Klein bottles in $ℝ4$ that have Euler normal number $− 4$ or $+ 4$ and Willmore energy $8π$. The surfaces are distinct even when we allow conformal transformations of $ℝ4$. As they are all minimizers in their regular homotopy class, they are Willmore surfaces.

Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2485-2526.

Dates
Revised: 8 August 2016
Accepted: 6 September 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.gt/1508437648

Digital Object Identifier
doi:10.2140/gt.2017.21.2485

Mathematical Reviews number (MathSciNet)
MR3654115

Zentralblatt MATH identifier
1365.53004

Keywords
Willmore surfaces Klein bottle

Citation

Breuning, Patrick; Hirsch, Jonas; Mäder-Baumdicker, Elena. Existence of minimizing Willmore Klein bottles in Euclidean four-space. Geom. Topol. 21 (2017), no. 4, 2485--2526. doi:10.2140/gt.2017.21.2485. https://projecteuclid.org/euclid.gt/1508437648

References

• L V Ahlfors, Complex analysis: an introduction of the theory of analytic functions of one complex variable, 2nd edition, McGraw-Hill, New York (1966)
• M F Atiyah, N J Hitchin, I M Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978) 425–461
• T F Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974) 407–413
• M Bauer, E Kuwert, Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. 2003 (2003) 553–576
• L P Eisenhart, Minimal surfaces in Euclidean four-space, Amer. J. Math. 34 (1912) 215–236
• J Eschenburg, R Tribuzy, Branch points of conformal mappings of surfaces, Math. Ann. 279 (1988) 621–633
• T Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984) 257–287
• J Hirsch, E Mäder-Baumdicker, A note on Willmore minimizing Klein bottles in Euclidean space, preprint (2016)
• M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242–276
• M Koecher, A Krieg, Elliptische Funktionen und Modulformen, revised edition, Springer (2007)
• R Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc. 17 (1987) 291–295
• 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” (B Chow, R Gulliver, S Levy, J Sullivan, editors), A K Peters, Wellesley, MA (1996) 67–72
• R Kusner, N Schmitt, The spinor representations of surfaces in space, preprint (1996)
• E Kuwert, Y Li, $W^{2,2}$–conformal immersions of a closed Riemann surface into $\mathbb R^n$, Comm. Anal. Geom. 20 (2012) 313–340
• E Kuwert, R Schätzle, Minimizers of the Willmore functional under fixed conformal class, J. Differential Geom. 93 (2013) 471–530
• H Lapointe, Spectral properties of bipolar minimal surfaces in $\mathbb S^4$, Differential Geom. Appl. 26 (2008) 9–22
• R Lashof, S Smale, On the immersion of manifolds in euclidean space, Ann. of Math. 68 (1958) 562–583
• H B Lawson, Jr, Complete minimal surfaces in $S\sp{3}$, Ann. of Math. 92 (1970) 335–374
• 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) 269–291
• F C Marques, A Neves, The Willmore conjecture, Jahresber. Dtsch. Math.-Ver. 116 (2014) 201–222
• W S Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969) 143–156
• J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
• R Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics 5, Amer. Math. Soc., Providence, RI (1995)
• T Rivière, Analysis aspects of Willmore surfaces, Invent. Math. 174 (2008) 1–45
• T Rivière, Variational principles for immersed surfaces with $L^2$–bounded second fundamental form, J. Reine Angew. Math. 695 (2014) 41–98
• S Salamon, Topics in four-dimensional Riemannian geometry, from “Geometry seminar `Luigi Bianchi”' (E Vesentini, editor), Lecture Notes in Math. 1022, Springer (1983) 33–124
• L Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993) 281–326
• H Whitney, The self-intersections of a smooth $n$–manifold in $2n$–space, Ann. of Math. 45 (1944) 220–246
• P Wintgen, Sur l'inégalité de Chen–Willmore, C. R. Acad. Sci. Paris Sér. A–B 288 (1979) A993–A995