109 research outputs found
Hypersurfaces with nonnegative Ricci curvature in hyperbolic space
Based on properties of n-subharmonic functions we show that a complete,
noncompact, properly embedded hypersurface with nonnegative Ricci curvature in
hyperbolic space has an asymptotic boundary at infinity of at most two points.
Moreover, the presence of two points in the asymptotic boundary is a rigidity
condition that forces the hypersurface to be an equidistant hypersurface about
a geodesic line in hyperbolic space. This gives an affirmative answer to the
question raised by Alexander and Currier in 1990.Comment: 14 page
Hypersurfaces in Hyperbolic Poincar\'e Manifolds and Conformally Invariant PDEs
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor
of a conformal representative of the conformal infinity of a hyperbolic
Poincar\'e manifold and the principal curvatures on the level sets of its
uniquely associated defining function with calculations based on [9] [10]. This
relationship generalizes the result for hypersurfaces in {\H}^{n+1} and their
connection to the conformal geometry of {\SS}^n as exhibited in [7] and gives
a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e
manifolds and conformally invariant equations on the conformal infinity. In
particular, we generalize an equivalence exhibited in [7] between
Christoffel-type problems for hypersurfaces in {\H}^{n+1} and scalar
curvature problems on the conformal infinity {\SS}^n to hyperbolic Poincar\'e
manifolds.Comment: 16 page
A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners Along a Hypersurface
In this paper we take an approach similar to that in [M] to establish a
positive mass theorem for asymptotically hyperbolic spin manifolds admitting
corners along a hypersurface. The main analysis uses an integral representation
of a solution to a perturbed eigenfunction equation to obtain an asymptotic
expansion of the solution in the right order. This allows us to understand the
change of the mass aspect of a conformal change of asymptotically hyperbolic
metrics
On nonnegatively curved hypersurfaces in hyperbolic space
In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded
Hypersurfaces with nonnegative Ricci curvature in hyperbolic space
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier (Proc Symp Pure Math 54(3):37â44, 1993)
Weakly horospherically convex hypersurfaces in hyperbolic space
In Bonini et al. (Adv Math 280:506â548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces Ï:MnâHn+1 and a class of conformal metrics on domains of the round sphere Sn . Some of the key aspects of the correspondence and its consequences have dimensional restrictions nâ„3 due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185â209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of Sn . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions nâ„2 in a unified way. In the case of a single point boundary ââÏ(M)={x}âSn , we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in H
Hypersurfaces in Hyperbolic Space with Support Function
Based on [previous publication*], we develop a global correspondence between immersed hypersurfaces Ï:MnâHn+1Ï:MnâHn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e2ÏgSne2ÏgSn on domains Ω in the boundary SnSn at infinity of Hn+1Hn+1, where Ï is the horospherical support function, ââÏ(Mn)=âΩââÏ(Mn)=âΩ, and Ω is the image of the Gauss map G:MnâSnG:MnâSn. To do so we first establish results on when the Gauss map G:MnâSnG:MnâSn is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hn+1Hn+1 and conformal metrics on domains in SnSn. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hn+1Hn+1 of constant mean curvature.
*J.M. Espinar, J.A. GĂĄlvez, P. Mira. Hypersurfaces in Hn+1Hn+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems. J. Eur. Math. Soc. (JEMS), 11 (2009), pp. 903â93
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