ﻻ يوجد ملخص باللغة العربية
We study the asymptotics as $puparrow 2$ of stationary $p$-harmonic maps $u_pin W^{1,p}(M,S^1)$ 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 $p_jto 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the support of a stationary, rectifiable $(n-2)$-varifold $V$ of density $Theta_{n-2}(|V|,cdot)geq 2pi$, given by the concentrated part of the measure $$mu=lim_{jtoinfty}(2-p_j)|du_{p_j}|^{p_j}dv_g.$$ When $n=2$, we show moreover that the density of $|V|$ takes values in $2pimathbb{N}$. Finally, on every compact manifold of dimension $ngeq 2$ we produce examples of nontrivial families $(1,2) i pmapsto u_pin W^{1,p}(M,S^1)$ of such maps via natural min-max constructions.
For a harmonic map $u:M^3to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2pi int_{thetain S^1}chi(Sigma_{theta})geq frac{1}{2}int_{thetain S^1}int_{Sigma_{theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M
We propose a new notion called emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $pto infty $.
Critical points of approximations of the Dirichlet energy `{a} la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energi
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [3
A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neigh