Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 815-842.

Global weak solutions of the Teichmüller harmonic map flow into general targets

Melanie Rupflin and Peter M. Topping

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Abstract

We analyse finite-time singularities of the Teichmüller harmonic map flow — a natural gradient flow of the harmonic map energy — and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.

Article information

Source
Anal. PDE, Volume 12, Number 3 (2019), 815-842.

Dates
Received: 19 December 2017
Accepted: 29 June 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1540432871

Digital Object Identifier
doi:10.2140/apde.2019.12.815

Mathematical Reviews number (MathSciNet)
MR3864211

Zentralblatt MATH identifier
06986454

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C43: Differential geometric aspects of harmonic maps [See also 58E20] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Keywords
geometric flows minimal surfaces harmonic maps

Citation

Rupflin, Melanie; Topping, Peter M. Global weak solutions of the Teichmüller harmonic map flow into general targets. Anal. PDE 12 (2019), no. 3, 815--842. doi:10.2140/apde.2019.12.815. https://projecteuclid.org/euclid.apde/1540432871


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References

  • B. Chow, “The Ricci flow on the $2$-sphere”, J. Differential Geom. 33:2 (1991), 325–334.
  • W. Ding and G. Tian, “Energy identity for a class of approximate harmonic maps from surfaces”, Comm. Anal. Geom. 3:3-4 (1995), 543–554.
  • W. Ding, J. Li, and Q. Liu, “Evolution of minimal torus in Riemannian manifolds”, Invent. Math. 165:2 (2006), 225–242.
  • J. Eells and L. Lemaire, “A report on harmonic maps”, Bull. London Math. Soc. 10:1 (1978), 1–68.
  • J. Eells, Jr. and J. H. Sampson, “Harmonic mappings of Riemannian manifolds”, Amer. J. Math. 86 (1964), 109–160.
  • G. Giesen and P. M. Topping, “Existence of Ricci flows of incomplete surfaces”, Comm. Partial Differential Equations 36:10 (2011), 1860–1880.
  • R. D. Gulliver, II, R. Osserman, and H. L. Royden, “A theory of branched immersions of surfaces”, Amer. J. Math. 95 (1973), 750–812.
  • R. S. Hamilton, “The Ricci flow on surfaces”, pp. 237–262 in Mathematics and general relativity (Santa Cruz, CA, 1986), edited by J. A. Isenberg, Contemp. Math. 71, Amer. Math. Soc., Providence, RI, 1988.
  • J. Hass and P. Scott, “The existence of least area surfaces in $3$-manifolds”, Trans. Amer. Math. Soc. 310:1 (1988), 87–114.
  • T. Huxol, M. Rupflin, and P. M. Topping, “Refined asymptotics of the Teichmüller harmonic map flow into general targets”, Calc. Var. Partial Differential Equations 55:4 (2016), art. id. 85.
  • W. Meeks, III, L. Simon, and S. T. Yau, “Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature”, Ann. of Math. $(2)$ 116:3 (1982), 621–659.
  • B. Randol, “Cylinders in Riemann surfaces”, Comment. Math. Helv. 54:1 (1979), 1–5.
  • C. Robertson and M. Rupflin, “Finite time degeneration for variants of Teichmüller harmonic map flow”, preprint, 2018.
  • M. Rupflin, “An improved uniqueness result for the harmonic map flow in two dimensions”, Calc. Var. Partial Differential Equations 33:3 (2008), 329–341.
  • M. Rupflin, “Flowing maps to minimal surfaces: existence and uniqueness of solutions”, Ann. Inst. H. Poincaré Anal. Non Linéaire 31:2 (2014), 349–368.
  • M. Rupflin and P. M. Topping, “Flowing maps to minimal surfaces”, Amer. J. Math. 138:4 (2016), 1095–1115.
  • M. Rupflin and P. M. Topping, “Horizontal curves of hyperbolic metrics”, Calc. Var. Partial Differential Equations 57:4 (2018), art. id. 106.
  • M. Rupflin and P. M. Topping, “Teichmüller harmonic map flow into nonpositively curved targets”, J. Differential Geom. 108:1 (2018), 135–184.
  • M. Rupflin, P. M. Topping, and M. Zhu, “Asymptotics of the Teichmüller harmonic map flow”, Adv. Math. 244 (2013), 874–893.
  • M. Struwe, “On the evolution of harmonic mappings of Riemannian surfaces”, Comment. Math. Helv. 60:4 (1985), 558–581.
  • P. Topping, “Winding behaviour of finite-time singularities of the harmonic map heat flow”, Math. Z. 247:2 (2004), 279–302.
  • P. M. Topping, “Uniqueness and nonuniqueness for Ricci flow on surfaces: reverse cusp singularities”, Int. Math. Res. Not. 2012:10 (2012), 2356–2376.
  • P. M. Topping and H. Yin, “Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces”, Ann. PDE 3:1 (2017), art. id. 6.