Nonexistence of quasi-harmonic spheres with large energy

Abstract

The absence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let (N , h) be a complete noncompact Riemannian manifold. Assume the universal covering of (N , h) admits a nonnegative strictly convex function with polynomial growth. Then there is no non-constant quasi-harmonic sphere u : R n → N such that lim r →∞ r n e − r 2 4 |x|≤r e − |x| 2 4 |∇u| 2 dx = 0.This generalizes a result of the first author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus comparatively elementary.

Liouville theorems for f-harmonic maps into Hadamard spaces

“…In particular, our f -harmonic functions lie in this setting. By applying Sturm's L p -Liouville theorem to f -harmonic functions, we immediately obtain several consequences which generalize previous results of [49,24,26,25]. Although the proof of L p -Liouville theorem is quite general which only involves the integration by parts and the Caccioppoli inequality (thus it holds for all reasonable spaces), it is surprisingly powerful to obtain various Liouville theorems for weighted manifolds with slow volume growth, especially for the Gaussian spaces, see Corollaries 2.2 and 2.3 in Section 2.…”

“…Analysis on weighted manifolds and the corresponding f -Laplacian have been extensively studied recently. We refer to [34,35,3,29,30] for the f -harmonic functions on weighted manifolds, to [24,49,26,25] for f -harmonic functions on the Gaussian spaces, to [12,13] for heat kernel estimates, and to [31,46,4,37,41,42] for f -harmonic maps.…”

Lorentzian harmonic map and its heat flow

“…Eells 和 Sampson 在关于存在性 的奠基性论文 [3] 中引入了热流的方法, 并且证明了当目标流形有非正截面曲率时, 每个同伦类中存 在调和映射; 随后, Hamilton [4] 和 Chang [5] 等把结论推广到带边流形的情形. 目标流形具有非正截面 曲率的条件可以进一步减弱, Ding 和 Lin [6] 证明了当目标流形的万有覆盖空间上存在平方增长的非 负严格凸函数时存在性成立; Li 和 Zhu [7] 及 Li 和 Yang [8] 把平方增长的条件减弱为多项式增长, 随后…”

Global existence of the harmonic map heat flow into Lorentzian manifolds

“…In the Riemannian case, such a result was first proved by Ding and Lin [11] when the universal covering of the target manifold admits a nonnegative strictly convex function with quadratic growth. The polynomial growth case was proved by Li-Zhu [28] and Li-Yang [27].…”