同伦映射类间的山路能量
Mountain pass energies between homotopy classes of maps
摘要：
对于闭黎曼流形之间的非同伦映射$u,v\in C^{\infty}(M,N)$，我们考虑存在连接$u_0=u$到$u_1=v$且具有$\|du_t\|_{L^p}^p\leq γ_p(u,v)$的路径$u_t\in W^{1,p}(M,N)$的最小能级$γ_p(u,v)$.当$u$和$v$是$(k-2)$ -同伦时，Hang和Lin的工作表明$γ_p(u,v)<\infty$对于$p\in [1,k)$，并且使用它们的构造，可以得到形式$γ_p(u,v)\leq \frac{C(u,v)}{k-p}$的估计。当$M$和$N$是定向的，$u$和$v$在$k-1$度的实上同调上诱导不同的映射时，我们证明了$γ_p(u,v)\sim \frac{1}{k-p}$作为$p\to k$的增长是尖锐的，并得到了系数$\liminf_{p\to k}(k-p)γ_p(u,v)$在$M$中的余维$k$整圈空间中某些不可收缩环的最小-最大质量的下界.在这个过程中，我们建立了一个相关的更小的量$γ_p^*(u,v)\leqγ_p(u,v)$的下界，对于该下界，存在满足$γ_p^*(u,v)\leq \|du_p\|_{L^p}^p\leq γ_p(u,v).$的$p$ -能量泛函的临界点$u_p\in W^{1,p}(M,N)$
英文摘要：
For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $γ_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq γ_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $γ_p(u,v)<\infty$ for $p\in [1,k)$, and using their construction, one can obtain an estimate of the form $γ_p(u,v)\leq \frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $γ_p(u,v)\sim \frac{1}{k-p}$ as $p\to k$ is sharp, and obtain a lower bound for the coefficient $\liminf_{p\to k}(k-p)γ_p(u,v)$ in terms of the min-max masses of certain non-contractible loops in the space of codimension-$k$ integral cycles in $M$. In the process, we establish lower bounds for a related smaller quantity $γ_p^*(u,v)\leqγ_p(u,v)$, for which there exist critical points $u_p\in W^{1,p}(M,N)$ of the $p$-energy functional satisfying $γ_p^*(u,v)\leq \|du_p\|_{L^p}^p\leq γ_p(u,v).$
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