Two-sided Hypersurfaces Research Articles

Two-sided hypersurfaces, entire Killing graphs and the mean curvature equation in warped products with density

In this paper, our purpose is to obtain uniqueness results related to the mean curvature equation for entire Killing graphs constructed over the base Pn of a warped product of the type Pfn×ρR with warping function ρ and density f. For this, we establish a suitable f-parabolicity criterion and, under appropriate constraints on the Bakry-Émery-Ricci tensor and on the f-mean curvature, we prove some rigidity results concerning two-sided hypersurfaces immersed in Pfn×ρR.

Large scale index of multi-partitioned manifolds

Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a Hermitean bundle on M with connection. We define a coarse multi-partitioned index of the spin Dirac operator on M twisted by E . Our main result is the computation of this multi-partitioned index as the Fredholm index of the Dirac operator on the compact manifold N , twisted by the restriction of E to N . We establish the following main application: if the scalar curvature of M is bounded below by a positive constant everywhere (or even if this happens only on one of the quadrants defined by the partitioning hypersurfaces) then the multi-partitioned index vanishes. Consequently, the multi-partitioned index is an obstruction to uniformly positive scalar curvature on M . The proof of the multi-partitioned index theorem proceeds in two steps: first we establish a strong new localization property of the multi-partitioned index which is the main novelty of this note. This we establish even when we twist with an arbitrary Hilbert A -module bundle E (for an auxiliary C^* -algebra A ). This allows to reduce to the case where M is the product of N with Euclidean space of dimension q . For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result.

Parabolicity and rigidity of two-sided hypersurfaces in Killing warped products

We extend a technique due to Romero et al. (Class Quantum Gav 30:1–13, 2013) establishing sufficient conditions to guarantee the parabolicity of a complete two-sided hypersurface immersed into a Killing warped product $$M\times _{\rho }{\mathbb {R}}$$ , whose base $$M^n$$ has parabolic universal Riemannian covering. Afterwards, we apply our parabolicity criterium in order to obtain a rigidity result concerning these hypersurfaces, whose mean curvature is not supposed a priori be constant. Finally, parametric uniqueness results are applied to obtain suitable non-parametric ones, i.e., to the case of entire Killing graphs.

Parabolicity and uniqueness of complete two-sided hypersurfaces immersed in a Riemannian warped product

In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type $$I\times _fM^n$$ , where $$M^{n}$$ is a connected n-dimensional oriented Riemannian manifold and $$f:I\rightarrow \mathbb {R}$$ is a positive smooth function. As applications, we obtain several uniqueness results concerning these hypersurfaces with constant mean curvature, under standard constraints on the Ricci curvature of $$M^n$$ and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity.

Liouville Type Results for Two-Sided Hypersurfaces in Weighted Killing Warped Products

We establish Liouville type results concerning two-sided hypersurfaces immersed in a weighted Killing warped product, under suitable constraints either on the Bakry-Emery-Ricci tensor of the base of the ambient space or on the height function of the hypersurface.

On the Uniqueness of Complete Two-Sided Hypersurfaces Immersed in a Class of Weighted Warped Products

Our aim in this work is to study the uniqueness of complete two-sided hypersurfaces immersed in a class of weighted warped product spaces, via application of some appropriate maximum principles. In particular, we obtain Moser–Bernstein type results concerning entire graphs in these ambient spaces.

Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product

Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product

Higher order mean curvature estimates for complete hypersurfaces into horoballs

We consider properly immersed two-sided hypersurfaces \({\varphi}:{M \rightarrow N}\) such that \({\varphi(M)}\) is contained in a horoball of N, where N satisfies fairly weak curvature bounds and we prove higher order mean curvature estimates that are natural extensions of the estimates obtained by Alias, Dajczer and Rigoli in [3] and Albanese, Alias and Rigoli in [1]. We show that these ambient curvature bounds in the presence of the properness of \({\varphi}\) guarantees that M satisfies a general version of the weak maximum principle established by Albanese, Alias and Rigoli in [1].

On Bernstein-type properties of complete hypersurfaces in weighted warped products

Through the application of appropriated generalized maximum principles, our aim in this work is to study Bernstein-type properties related to complete two-sided hypersurfaces immersed in a weighted warped product space. In this setting, supposing a natural comparison inequality between the weighted mean curvatures of the hypersurface and those of the slices of the slab where the hypersurface is supposed to be contained, we establish sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, we also obtain several rigidity results concerning the slices of weighted product spaces.

On the angle of complete CMC hypersurfaces in Riemannian product spaces

We deal with complete two-sided hypersurfaces immersed with constant mean curvature in a Riemannian product space R×Mn. First, when the fiber Mn is compact with positive sectional curvature, we apply Bochner's technique jointly with Omori–Yau's generalized maximum principle to establish sufficient conditions which assure that such a hypersurface Σn is a slice {t}×Mn, provided that its (not necessarily constant) angle has some suitable boundedness. Afterwards, when Mn is complete with nonnegative sectional curvature, we proceed with our technique in order to guarantee that a complete constant mean curvature hypersurface immersed in R×Mn is minimal. Moreover, we characterize the uniqueness of entire vertical graphs Σn(u) with constant mean curvature into R×Mn, in terms of the norm of the gradient of the function u which determines such a graph. A nontrivial example of entire vertical graph with constant angle in R×H2 is also given, showing that our results do not hold when the fiber of the ambient product space has negative sectional curvature.

On the scalar curvature of hypersurfaces in spaces with a Killing field

Abstract We consider compact hypersurfaces in an (n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when n = 2 and is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ≥ 3 and is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product 𝕄 × ℝ1.