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.
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.
Reviewer: Peter B. Gilkey (Eugene)
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 |