Conformal spectrum and harmonic maps (Q2801870)

Let \((M,g)\) be a closed connected smooth Riemannian surface of area 1 and \(\lambda_1(g)\) the first positive eigenvalue of its Laplace-Beltrami operator \(\Delta_g\). Let \([g]\) be the (singular) conformal class of \(g\) containing the metrics \(g_\mu=\mu g\) with area 1 and where \(\mu\in L^1(M,g)\). Let \(\Lambda_1(M,[g])=\sup_{h\in[g]}\lambda_1(h)\). \textit{J. Hersch} [C. R. Acad. Sci., Paris, Sér. A 270, 1645--1648 (1970; Zbl 0224.73083)] proved in 1970 the maximal bound \(\lambda_1(g)\leq 8\pi\) for any metric \(g\) on the sphere while \textit{P. C. Yang} and \textit{S.-T. Yau} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Sér. 7, 55--61 (1980; Zbl 0446.58017)] proved later the (non maximal) bound \(\lambda_1(g)\leq 8\pi[(\gamma+3)/2]\) for any surface \((M,g)\) with genus \(\gamma\).NEWLINENEWLINEThe authors prove here that, under the hypothesis \(\Lambda_1(M,[g])>8\pi\), there exists an extremal metric \(g_\mu\in[g]\) such that \(\Lambda_1(M,[g])=\lambda_1(g_\mu)\) with \(\mu\) a positive function outside of a finite number of points, where the metric has conical singularities. Moreover, there exists a family of eigenvectors \((u_1,\dots,u_\ell)\) with eigenvalue \(\lambda_1(g_\mu)\) such that the map \(\phi: m\in M\to (u_1(m),\dots,u_\ell(m))\in\mathbb{R}^{\ell}\) is an harmonic map on \(M\) with values in the sphere \(\mathbb S^{\ell-1}\); if \(\ell>2\), the map \(\phi:(M,g_\mu)\to (\mathbb S^{\ell-1},g_{\mathrm{round}})\) is a minimal branched conformal immersion.NEWLINENEWLINEThe proof is based on the analysis of the solutions \(\mu\) (\textit{a priori} not necessary non negative) for the Schrödinger equation \(\Delta_gu=\lambda_1(\mu g)\mu u\) on \(M\) with the constraint \(\int_M\mu dA_g=1\). Taking a maximalizing sequence \((\mu_k dA_g)\) of probability measures with density obeying the bound \(\mu_k(M)\subset[-1/2,k]\), the authors estimate carefully the subsets where \(\mu\) is non positive: after the proof of \textit{a priori} regularity results, they pass finally to the limit.NEWLINENEWLINELet us quote works where similar problems are considered: \textit{G. Kokarev} [Adv. Math. 258, 191--239 (2014; Zbl 1296.58020)] gives results complementary to this paper and \textit{R. Petrides} [Geom. Funct. Anal. 24, No. 4, 1336--1376 (2014; Zbl 1310.58022)] proves the hypothesis \(\Lambda_1(M,[g])>8\pi\) for any surface of positive genus before establishing the existence of extremal metrics.