Abstract The main goal of this paper is to present a complete description of all translation hypersurfaces with constant r-curvature S r, in the Euclidean space ℝ n + 1, where 3 ≤ r ≤ n - 1.

Estabelecemos um método para obter limites inferiores para o tom fundamental de operadores elípticos em forma divergente em termos do divergente de campos de vetores. Aplicamos esse método para os operadores Lr associados a hipersuperfícies imersas nas formas espaciais Nn(c) de curvatura seccional constante c com (r + 1)-curvatura média Hr + 1 localmente limitada. Obtemos como corolário limites inferiores para o raio extrínseco de hipersuperfícies compactas das formas espaciais Nn(c) com Hr + 1 > 0 em termos das r-ésima e r + 1-ésima curvatura médias. Finalmente, observamos que limites para os autovalores do Laplaciano essencialmente limitam os autovalores dos operadores elípticos em forma divergente. Isso permite mostrar que a constante de Cheeger limita inferiormente o primeiro autovalor não-nulo dos operadores Lr em hypersuperfícies compactas de Nn(c).

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).