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Register a new user$p$-Harmonic Maps to $S^1$ and Stationary Varifolds of Codimension 2
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Daniel Stern
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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.
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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 $R_M$ of $M$ to the average Euler characteristic of the level sets $Sigma_{theta}=u^{-1}{theta}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;mathbb{Z})$ in terms of $|R_M^-|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(min R_M)sys_2(M)leq 8pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.
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