Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $Φ(f)$ $=$ ${\displaystyle \int_M |T_f|^2dv_g}$. We call such maps {\it C-stationary maps}. Any conformal map or more generally any weakly conformal map is a C-stationary map. It is of interest to find when a C-stationary map is a (weakly) conformal map.
In this paper we prove the following result. If $f$ is a stable C-stationary maps from the standard sphere ${\mathrm S}^m$ $(m \geq 5)$ or into the standard sphere ${\mathrm S}^n$ $(n \geq 5)$, then $f$ is a weakly conformal map.
Let $(M, J^α, α=1,2,3)$ and $(N, {\cal J}^α, α=1,2,3)$ be Hyperkähler manifolds. Suppose that $u_k$ is a sequence of stationary quaternionic maps and converges weakly to $u$ in $H^{1,2}(M,N)$, we derive a blow-up formula for $\lim_{k\to\infty}d(u_k^*{\cal J}^α)$, for $α=1,2,3$, in the weak sense. As a corollary, we show that the maps constructed by Chen-Li [CL2] and by Foscolo [F] can not be tangent maps (c.f [LT], Theorem 3.1) of a stationary quaternionic map satisfing $d(u^*{\cal J}^α)=0$.
Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto's theorem to complete space-like stationary surfaces in Minkowski spacetime, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in 4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish …
We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $$\int_M|du_p|^p=O(\frac{1}{2-p}).$$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the support of a stationary, rectifiable $(n-2)$-varifold $V$ of density $Θ_{n-2}(\|V\|,\cdot)\geq 2π$, given by the concentrated part of the measure $$μ=\lim_{j\to\infty}(2-p_j)|du_{p_j}|^{p_j}dv_g.$$ When $n=2$, we show moreover that the density of $\|V\|$ takes values in $2π\mathbb{N…
Energy identity for stationary biharmonic mappings into spheres in supercritical dimensions
Shandong University（山东大学）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education, P. R. China（非线性期望前沿科学中心，中华人民共和国教育部）；University of Eastern Finland（芬兰东部大学）；Purdue University（普渡大学）；China Three Gorges University（三峡大学）
本文研究了黎曼流形之间极小化和平稳$p$ -调和映射的正则性。我们的目的是获得Minkowski型的体积估计的奇异集$S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$，而不是较弱的和非定量的豪斯多夫维数界目前在文献中通用的$p$。本文的主要技术是基于对$f$切映射的近似对称性的研究的定量分层。在这篇文章中，我们推广了在\cite{ChNa2}进行的研究，以最小化$2$ -调和映射到一般的$p\in (1,\infty)$。此外，我们还分析了静态的情况下，缺乏紧凑性，使研究更加复杂。为了理解退化固有的平稳映射的行为，我们研究自然相关的一系列这样的地图和推广的结果在\cite{lin_stat}的缺陷措施。利用精化覆盖的方法，改进了孤立奇点情形下的估计，得到了奇点个数的一个确定界。即使对于最小化$2$ -调和映射，这个结果似乎也是新的。
Energy identity for Intrinsic Stationary Biharmonic Mappings into Homogeneous Spaces in Supercritical Dimensions
Shandong University（山东大学）；University of Eastern Finland（芬兰东部大学）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education, P. R. China（教育部非线性期望前沿科学中心）；Tsinghua University（清华大学）
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
