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Đề tài " Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow "

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We present an analysis of bounded-energy low-tension maps between 2-spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop, we show that we can establish a ‘quantization estimate’ which constrains the energy of the map to lie near to a discrete energy spectrum. One application is to the asymptotics of the harmonic map flow; we find uniform exponential convergence in time, in the case under consideration. | Annals of Mathematics Repulsion and quantization in almost-harmonic maps and asymptotics of the harmonic map flow By Peter Topping Annals of Mathematics 159 2004 465 534 Repulsion and quantization in almost-harmonic maps and asymptotics of the harmonic map flow By Peter Topping Abstract We present an analysis of bounded-energy low-tension maps between 2-spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop we show that we can establish a quantization estimate which constrains the energy of the map to lie near to a discrete energy spectrum. One application is to the asymptotics of the harmonic map flow we find uniform exponential convergence in time in the case under consideration. Contents 1. Introduction 1.1. Overview 1.2. Statement of the results 1.2.1. Almost-harmonic map results 1.2.2. Heat flow results 1.3. Heuristics of the proof of Theorem 1.2 2. Almost-harmonic maps the proof of Theorem 1.2 2.1. Basic technology 2.1.1. An integral representation for eg 2.1.2. Riesz potential estimates 2.1.3. Lp estimates for ed and eg 2.1.4. Hopf differential estimates 2.2. Neck analysis 2.3. Consequences of Theorem 1.1 2.4. Repulsive effects 2.4.1. Lower bound for ed off T-small sets 2.4.2. Bubble concentration estimates Partly supported by an EPSRC Advanced Research Fellowship. 466 PETER TOPPING 2.5. Quantization effects 2.5.1. Control of e.Q 2.5.2. Analysis of neighbourhoods of antiholomorphic bubbles 2.5.3. Neck surgery and energy quantization 2.5.4. Assembly of the proof of Theorem 1.2 з. Heat flow the proof of Theorem 1.7 1. Introduction 1.1. Overview. To a sufficiently regular map u S2 - S2 R3 we may assign an energy 1.1 E u 1 Vu 2 2 J S2 and a tension field 1.2 T u Au u Vu 2 orthogonal to u which is the negation of the L2-gradient of the energy E at и. Critical points of the energy i.e. maps u for which T u 0 are called harmonic maps. In this situation the harmonic maps are precisely the rational maps .

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