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For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.

Let Ω be a bounded domain in an n-dimensional Euclidean space R n . We study eigenvalues of an eigenvalue problem of a system of elliptic equations:Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the (k + 1) th eigenvalue σ k+1 . We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature.Key words and phrases: Universal bounds, eigenvalues, a system of elliptic equations, Cheng-Yang's inequality, biharmonic operator, positive Ricci curvature.

Treatment of spontaneously hypertensive rats (SHR) with captopril (100 mg ⋅ kg−1 ⋅ day−1) throughout development and during the first 16 wk of life leads to a reduction in blood pressure and left ventricular hypertrophy. Blood pressures and hypertrophy are reduced in these animals (vs. untreated SHR) for up to 24 wk after discontinuation of the drug. We used conventional blot hybridization and Western analysis to examine hypertrophy-dependent gene expression during this period. Ventricular expression of the atrial natriuretic peptide gene was reduced by >90% at 16 wk of age in the captopril-treated SHR. Expression increased in the 24 wk after discontinuation of treatment, but remained well below that of the untreated SHR. A similar reduction in ventricular c- myc gene expression was seen with captopril treatment. Neither renal expression of the atrial natriuretic peptide gene nor ventricular expression of the c- fos gene was affected by captopril. This study demonstrates that captopril treatment during a critical period of development in the SHR leads to a sustained reduction in hypertrophy-dependent myocardial gene expression, which does not revert to levels seen in the untreated SHR after discontinuation of the drug.

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal if this conjecture holds, obviously. In this paper, using new method, we solve this conjecture in the Euclidean space case intrinsically. We think our method is valid for the case of general Riemannian manifolds and give some examples directly.Comment: 15 page

Given a compact Riemannian manifold M, we consider a warped productM = I × h M where I is an open interval in R. For a positive function ψ defined onM, we generalized the arguments in [28] and [42], to obtain the curvature estimates for Hessian equations σ k (κ) = ψ(V, ν(V)). We also obtain some existence results for the starshaped compact hypersurface Σ satisfying the above equation with various assumptions.2010 Mathematics Subject Classification. Primary 53C45, Secondary 53J60.

For a bounded planar region in R 2 , we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k + 1)-th (k 3) membrane eigenvalues.