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

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 . Overview . Statement of the results . Almost-harmonic map results . Heat flow results . Heuristics of the proof of Theorem 2. Almost-harmonic maps the proof of Theorem . Basic technology . An integral representation for eg . Riesz potential estimates . Lp estimates for ed and eg . Hopf differential estimates . Neck analysis . Consequences of Theorem . Repulsive effects . Lower bound for ed off T-small sets . Bubble concentration estimates Partly supported by an EPSRC Advanced Research Fellowship. 466 PETER TOPPING . Quantization effects . Control of . Analysis of neighbourhoods of antiholomorphic bubbles . Neck surgery and energy quantization . Assembly of the proof of Theorem з. Heat flow the proof of Theorem 1. Introduction . Overview. To a sufficiently regular map u S2 - S2 R3 we may assign an energy E u 1 Vu 2 2 J S2 and a tension field 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 . maps u for which T u 0 are called harmonic maps. In this situation the harmonic maps are precisely the rational maps .