0000000000336215
AUTHOR
M. Carmen Domingo-juan
Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF
Let $(\bar{M},,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{{\psi}} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{{\psi}}$ is the mean curvature of $M$ associated to the density $e^\psi$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian den…
Pappus type theorems for hypersurfaces in a space form
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM λ n of sectional curvature λ. LetP 0 be a totally geodesic hypersurface ofM λ n throughc(0) and orthogonal toc(t). LetC 0 be a hypersurface ofP 0. LetC be the hypersurface ofM λ n obtained by a motion ofC 0 alongc(t). We shall denote it byC PorC Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC 0, then volume(C) ≥ volume(C),P),and the equality holds whenC 0 is contained in a geodesic sphere or…
Pappus type theorems for motions along a submanifold
Abstract We study the volumes volume( D ) of a domain D and volume( C ) of a hypersurface C obtained by a motion along a submanifold P of a space form M n λ . We show: (a) volume( D ) depends only on the second fundamental form of P , whereas volume( C ) depends on all the i th fundamental forms of P , (b) when the domain that we move D 0 has its q -centre of mass on P , volume( D ) does not depend on the mean curvature of P , (c) when D 0 is q -symmetric, volume( D ) depends only on the intrinsic curvature tensor of P ; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO ( n − q − d ), and C …