Geometry & Topology

Existence of minimizing Willmore Klein bottles in Euclidean four-space

Patrick Breuning, Jonas Hirsch, and Elena Mäder-Baumdicker

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Let K = P2 P2 be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles f : K n 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 4, 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 4 that have Euler normal number 4 or + 4 and Willmore energy 8π. The surfaces are distinct even when we allow conformal transformations of 4. As they are all minimizers in their regular homotopy class, they are Willmore surfaces.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C28: Twistor methods [See also 32L25] 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space

Willmore surfaces Klein bottle


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.

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  • 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