Mathematics – Differential Geometry
Scientific paper
2012-03-31
Mathematics
Differential Geometry
17 pages. Comments are welcome
Scientific paper
Let $\Phi:TM\to TM$ be a positive-semidefinite operator of class $C^1$ defined on a complete noncompact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $\Phi$ and on the second fundamental form to guarantee that either $\Phi\equiv 0$ or the integral $\int_M trace\Phi dM$ is infinite. Some applications are given here. One of them says that if $M$ admits an integrable distribution whose integrals are minimal submanifolds in $\bar{M}$ then the volume of $M$ must be infinite. Another one says that if the sectional curvature of $\bar{M}$ satisfies $\bar{K}\leq -c^2$, for some $c\geq 0$, and $\lambda:M\to [0,\infty)$ is a nonnegative $C^1$ function such that gradient vector of $\lambda$ and the mean curvature vector $H$ of the immersion satisfy $|H+p\nabla\lambda|\leq c \lambda$, for some $p\geq 1$, then either $\lambda\equiv 0$ or the integral $\int_M {\lambda}^p dM$ is infinite.
Batista Màrcio
Mirandola Heudson
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