Analytic Aspects of the Harmonic Map Problem

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

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Harmonic branched coverings and uniformization of CAT(k) spheres

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0031 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Christine Breiner ◽

Chikako Mese

Keyword(s):

Harmonic Map ◽

Branched Coverings ◽

Curvature Bound ◽

Branched Covering

Abstract Let S be a surface with a metric d satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into ( S , d ) {(S,d)} is a branched covering. As a consequence, if ( S , d ) {(S,d)} is homeomorphically equivalent to the 2-sphere 𝕊 2 {\mathbb{S}^{2}} , then it is conformally equivalent to 𝕊 2 {\mathbb{S}^{2}} .

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Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

Advances in Calculus of Variations ◽

10.1515/acv-2019-0086 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

James Kohout ◽

Melanie Rupflin ◽

Peter M. Topping

Keyword(s):

Constant Curvature ◽

Harmonic Map ◽

Gradient Flow ◽

Minimal Immersion ◽

Curvature Surface ◽

Previous Theory ◽

Target Manifold ◽

Harmonic Map Flow

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

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Transition of blow-up mechanisms in k-equivariant harmonic map heat flow

Nonlinearity ◽

10.1088/1361-6544/ab74f4 ◽

2020 ◽

Vol 33 (6) ◽

pp. 2756-2796

Author(s):

Paweł Biernat ◽

Yukihiro Seki

Keyword(s):

Heat Flow ◽

Blow Up ◽

Harmonic Map ◽

Harmonic Map Heat Flow

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Some results on stable p-harmonic maps

Glasgow Mathematical Journal ◽

10.1017/s0017089500030561 ◽

1994 ◽

Vol 36 (1) ◽

pp. 77-80 ◽

Cited By ~ 5

Author(s):

Leung-Fu Cheung ◽

Pui-Fai Leung

Keyword(s):

Riemannian Manifold ◽

Critical Point ◽

Harmonic Maps ◽

Harmonic Map ◽

Energy Functional ◽

Complete Riemannian Manifold ◽

Image Position ◽

Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

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A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

Stochastic Processes and their Applications ◽

10.1016/j.spa.2014.06.004 ◽

2014 ◽

Vol 124 (11) ◽

pp. 3535-3552 ◽

Cited By ~ 4

Author(s):

Hongxin Guo ◽

Robert Philipowski ◽

Anton Thalmaier

Keyword(s):

Heat Flow ◽

Harmonic Map ◽

Stochastic Approach ◽

Riemannian Metric ◽

Time Dependent ◽

Harmonic Map Heat Flow

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BOCHNER-TYPE FORMULAS FOR TRANSVERSALLY HARMONIC MAPS

International Journal of Mathematics ◽

10.1142/s0129167x11007471 ◽

2012 ◽

Vol 23 (03) ◽

pp. 1250003 ◽

Cited By ~ 1

Author(s):

QUN CHEN ◽

WUBIN ZHOU

Keyword(s):

Harmonic Maps ◽

Harmonic Map ◽

Transverse Curvature ◽

Sasaki Manifold ◽

Sasaki Manifolds

The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M′, and M′ has a strongly negative transverse curvature. If the rank of dTf is at least three at some points of M, then f is contact holomorphic (or contact anti-holomorphic).

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