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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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