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Riemannian manifolds admitting isometric immersions by their first eigenfunctions. (English) Zbl 1030.53043
Let \(M\) be a compact smooth manifold of dimension \(m\geq 2\). Denote by \({\mathcal R}_0(M)\) the set of Riemannian metrics of volume 1 on \(M\). For any \(g\in {\mathcal R}_0 (M)\), denote by \(0<\lambda_1 (g)\leq\lambda-2(g)\leq \cdots \leq \lambda_k(g) \leq\cdots\) the increasing sequence of eigenvalues of the Laplacian \(\Delta_g\) of \(g\). The functional \[ \lambda_1: {\mathcal R}_0(M)\to \mathbb{R}, \] \[ g\mapsto \lambda_1(g) \] is continuous but not differentiable in general. However, for any family \((g_t)_t\) of metrics, analytic in \(t\), \(\lambda_1 (g_t)\) has right and left derivatives w.r.t. \(t\).
A metric \(g\in {\mathcal R}(M)\) is said to be extremal for the \(\lambda_1\) functional if for any analytic deformation \((g_t)_t\subset {\mathcal R}_0(M)\), with \(g_0=g\), the left and right derivatives of \(\lambda_1(g_t)\) at \(t=0\) have opposite signs, i.e. \[ {d \over dt}\lambda_1 (g_t)|_{t=0^+} \leq 0\leq {d\over dt}\lambda_1 (g_t) |_{t=0^-}. \] A metric \(g\) on a compact \(m\)-dimensional manifold \(M\) is \(\lambda_1\)-minimal if the eigenspace \(E_1(g)\) associated with the first nonzero eigenvalue \(\lambda_1(g)\) of the Laplacian of \(g\) contains a family \(f_1, \dots, f_k\) of functions satisfying: \(\sum_{1\leq i\leq k} df_i\otimes df_i=g\). It follows from a well known result of Takahashi that this last condition is equivalent to the fact that the map \(f=(f_1,\dots,f_k)\) is a minimal isometric immersion from \((M, g)\) into the Euclidean sphere \(S_r^{k-1}\) of radius \(r=\sqrt {m\over \lambda_1 (g)}\).
Then the main result of the authors is
Theorem 1: If a Riemannian metric \(g\in {\mathcal R}_0(M)\) is extremal for \(\lambda_1\) then it is \(\lambda_1\)-minimal.
They prove a certain converse of Theorem 1.
Theorem 2: Let \(g\in {\mathcal R}_0(M)\) and assume there exists an \(L_2(g)\)-orthonormal basis \(\{\varphi_1, \dots, \varphi_k\}\) of \(E_1(g)\) such that 2-tensor \(\sum_{1<i<k} d\varphi_i \otimes d\varphi_i\) is proportional to \(g\). Then \(g\) is extremal for \(\lambda_1\).
Finally they present \(\lambda_1\)-minimal and extremal metrics on the torus.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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