Regularity of minimizing harmonic mappings into ellipsoids and similar other manifolds (Q1803886)

Suppose \(N\) is an \(n(\geq 3)\)-dimensional Riemannian manifold which satisfies the condition: there exist \(r>0\) and \(p\in N\) such that (i) On the closed ball \(B(p,r)\) in \(N\) there is a normal coordinate system \((y^ 1,\dots,y^ n)\) with \(p=(0,\dots,0)\); (ii) For any normalized geodesic \(\sigma\) with \(\sigma(0)=p\) and any nontrivial Jacobi field \(Y\) along \(\sigma\) with \(Y(t)\perp\dot\sigma(t)\) and \(Y(0)=0\) one has \({d\over dt} | Y(t) |^ 2>0\) for \(t \in(0,r)\) and \({d \over dt} | Y(t) |^ 2=0\) for \(t=r\). Let \(K_ y(\xi,\eta)\) be the sectional curvature of \(N\). Define \(k(y)=\min\{K_ y(\xi_ 1,\xi_ 2):\xi_ 1=\dot\sigma(1),\xi_ 2\in T_ yN\}\), \(K(y)=\max\{K_ y(\xi_ 1,\xi_ 2):\xi_ 1,\xi_ 2 \in T_ yN\) l.i., \(\xi_ 1,\xi_ 2\perp\dot \sigma(1)\}\), here \(\sigma: [0,1]\to N\) is the unique geodesic in \(B(p,r)\) joining \(p\) to \(y\in N\). Let \(L=\min_{y \in \partial B(p,r)}(k(y)/K(y))\) and \[ d=\begin{cases} 2 \quad\text{if } L<(2n- 2)/(4n-3) \\ 3\quad\text{if } L\in[(2n-2)/(4n-3),(4n-4)/(6n-5)) \\ 4 \quad \text{if } L\in[(4n-4)/(6n-5),(6n-6)/(8n-7)) \\ 5\quad \text{if } L\in[(6n-6)/(8n-7),4/5] \\ l\quad\text{if } l\geq 6\text{ and } L\in((l- 2)^ 2/4(l-1),(l-1)^ 2/4l]. \end{cases} \] Assume \(K(y)>0\) for all \(y \in \partial B(p,r)\). Let \(M\) be an \(m(\geq 3)\)-dimensional compact connected Riemannian manifold. By using the regularity theory of \textit{R. Schoen} and \textit{K. Uhlenbeck} [J. Differ. Geom. 17, 307-335; correction, ibid. 19, 329 (1982; Zbl 0521.58021)] and the derivation of a stability inequality from the formula for the second variation of energy, the author proves the following results: Every locally energy minimizing mapping \(u:M\to\overline{B(p,r)}\subseteq N\) is regular on \(M\backslash \partial M\) if \(3\leq m\leq d\); if \(m\geq d+1\) then \(u\) is regular on \(M\backslash \partial M\) outside a closed set \({\mathcal S}\) with the Hausdorff dimension of \({\mathcal S}\leq m-d-1\); if \(m=d+1\), then \({\mathcal S}\) is discrete. Besides, the curvature condition \(L>(l-2)^ 2/4(l-1)\) which implies \({\mathcal H}\)-dim\(({\mathcal S})\leq m-l-1\) is optimal for \(l\geq 7\).