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Extrinsic upper bound for the first eigenvalue of elliptic operators. (English) Zbl 1085.58024

Let \((M,g)\) be a compact, connected, smooth \(m\)-dimensional Riemannian manifold. Let \(T\) be a positive definite symmetric divergence free \(1-1\) tensor. Let \(L_T(u):=-\text{div}(T\nabla u)\). If \(T\) is the identity, then \(L_T\) is just the usual scalar Laplacian. Let \(\lambda_1(T)\) be the first eigenvalue of this elliptic operator. Let \(\phi\) be an isometric immersion of \((M,g)\) into an \(n\)-dimensional complete Riemannian manifold \((N,h)\) of sectional curvature bounded above by \(\delta\). If \(\delta\leq0\), assume \((N,h)\) is simply connected. If \(\delta>0\), assume \(\phi(M)\) is contained in a convex ball of radius less than or equal to \(\pi/4\sqrt{\delta}\). Let \(B\) be the second fundamental form and let \(H_T(x)=\sum_iB(Te_i,e_i)\). The author shows:
Theorem. Adopt the notation established above. One has \[ \lambda_1(T)\leq\{\sup_M| H_T| ^2+\sup_M\delta(\text{tr}(T))^2\}/\{\inf_M\text{tr}(T)\}\,. \]
Additional estimates are obtained in different geometric contexts as well.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P15 Estimates of eigenvalues in context of PDEs
49R50 Variational methods for eigenvalues of operators (MSC2000)
31C12 Potential theory on Riemannian manifolds and other spaces
53C24 Rigidity results
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