Min-max harmonic maps and a new characterization of conformal eigenvalues (Q6620332)

Let \((M,g)\) be a closed Riemannian surface. The quantity \(\overline{\lambda}_{k}(M,g)= \lambda_{k}(M,g)\cdot Area(M,g)\) is scaling invariant, \(k=0,1,\dots\), where \[0=\lambda_{0}(M,g)<\lambda_{1}(M,g)\leq \lambda_{2}(M,g)\leq \cdots\to \infty\] are the Laplace eigenvalues. Given a conformal class \(c=[g]=\{f\cdot g, f: M \to (0,\infty)\}\) of the metric \(g\) consider \[\Lambda_k(M,c)=\sup_{h\in c}\overline{ \lambda}_{k}(M,h).\]\N\NThe metric realizing \(\Lambda_k(M,c)\), i.e., \(\Lambda_k(M,c)=\overline{ \lambda}_{k}(M,h)\) is related to harmonic maps \(\Phi: (M,g)\to \mathbb{S}^{n}\) through the following result: There exists a harmonic map \[\Phi: (M,g)\to \mathbb{S}^{n}\] such that the components of \(\Phi\) are \(\lambda_{k}(M,h)\)-eigenfunctions (see [\textit{A. El Soufi} and \textit{S. Ilias}, J. Geom. Phys. 58, No. 1, 89--104 (2008; Zbl 1137.49040); \textit{A. Fraser} and \textit{R. Schoen}, Contemp. Math. 599, 105--121 (2013; Zbl 1321.35118); \textit{N. Nadirashvili}, Geom. Funct. Anal. 6, No. 5, 877--897 (1996; Zbl 0868.58079)]).\N\NIn this paper the authors characterize \(\Lambda_1(M,c)\) and \(\Lambda_2(M,c)\) as the min-max energies associated to certain families of sphere valued maps on \(M\). Precisely, they prove the following results:\N\begin{itemize}\N\item[(1)] Theorem 1.2. For any closed Riemannian surface \((M,g)\) there exists a harmonic map \(\Psi_n : M \to \mathbb{S}^{n}\), \(n>5\) such that \(E(\Psi_n)\geq \frac{1}{2}\Lambda_1(M,c)\) whose energy index \(Ind_{E}(\Psi_n)\leq n+1\).Here \(E(\Psi_n)\) is the Dirichlet energy of the map.\N\item[(2)] Theorem 1.3. Given a conformal class of \(c=[g]\) there exists a number \(N=N(M, c)\) such that for all \(n\geq N\), the components of \(\Psi_n\) lie in the first positive eigenspaces of the Laplacian \(\Delta_{g_{\Psi_n}}\) for the conformal metric \(g_{\Psi_n}=\vert d \Psi_n\vert_{g}^2 \cdot g\) (it may have conical singularities). In particular, \(E(\Psi_n)= \frac{1}{2}\Lambda_{1}(M,c)\).\N\end{itemize}\N\NThen few applications of these results are derived, one of which is the following.\N\begin{itemize}\N\item[(3)] Let \(\Omega \subset M\) be a Lipschitz domain. Then \(\sigma_1(\Omega. g) Length(\partial \Omega, g) < \Lambda_1(M, g)\), where \(\sigma_1(\Omega, g)\) is the first Steklov eigenvalue.\N\end{itemize}\NThe paper has other similar results regarding \(\Lambda_2(\Omega. c)\).