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Register a new userComparison principle, stochastic completeness and half-space theorems
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We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in $mathbb{H}^n times mathbb{R}^ell$.
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We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $mathbb{R}^{3}_{raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $Mtimes R_+$ equals some $Mtimes {s}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $text{Ric}_M geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $R^3$ with bounded curvature if $sup vert H_{_M}vert < H$, or ${rm dist}(M,N)=0$ when $sup vert H_{_M}vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.
We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.
In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.
In this paper, we establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, we prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, we give some applications of these generalized maximum principles.
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a in (0, 1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a abla u)=0$ on the upper half space.
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