Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces
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Calculus of Variations
Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces Christine Breiner1 · Sajjad Lakzian2,3 Received: 16 April 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost (Two-dimensional geometric variational problems. Pure and applied mathematics. Wiley, New York, 1991) and Parker (J Differ Geom 44(3):595–633, 1996) respectively established these results by exploiting now classical arguments for harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. In the absence of a PDE, we take advantage of the local convexity properties of the target space. Included in this paper are an -regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image. Mathematics Subject Classification 53C43 · 58E20
1 Introduction In pioneering work, Sacks and Uhlenbeck [15] determined a priori estimates for critical points to a perturbed energy functional to prove the existence of minimal two-spheres in compact Riemannian manifolds. Recently, Breiner et al. [2] extended this result to the singular setting.
Communicated by R. Schoen. CB was supported in part by NSF Grant DMS-1609198.
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Christine Breiner [email protected] Sajjad Lakzian [email protected]; [email protected]
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Department of Mathematics, Fordham University, Bronx, NY 10458, USA
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School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
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Department of Mathematical Sciences, Isfahan University of Technology (IUT), Isfahan 8415831111, Iran 0123456789().: V,-vol
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C. Breiner and S. Lakzian
Lacking a PDE, they instead used the local convexity of the target space (a locally CAT(1) space) to determine a discrete harmonic map heat flow or harmonic replacement process. Given a finite energy map φ : (M 2 , g) → (X , d), harmonic replacement yields either a harmonic map u : (M 2 , g) → (X , d), homotopic to φ, or a conformal harmonic map v : (S2 , g0 ) → (X , d). The second case occurs when the modulus of continuity for the sequence of replacement maps blows up at a point. Renormalizing the domain on the scale of the blow up gives a sequence of finite energy maps with uniform modulus of continuity, which converge in C 0 uniformly on compact sets to the map v : C → X . By proving a removable singularity theorem for conformal harmonic maps, in [2] they concluded that v is harmonic on S2 . In the smooth setting, Sacks and Uhlenbeck [15] proved a removable singularity theorem for harmonic maps; coupling this result with their a p
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