Geodesic Hypersurfaces Research Articles

Some connections between symmetry results for semilinear PDE in real and hyperbolic spaces

Taking advantage of the “invariance” under conformal transformations of certain elliptic operators and combining it with symmetry results obtained by moving totally geodesic hypersurfaces in H n , we are able to prove the symmetry of positive solutions of − Δ u = f ( r , u ) , in balls in R n , for a class of nonlinearities that do not satisfy the classical hypothesis of f being decreasing in r.

Conformal Killing horizons

For time dependent black hole space–times the event horizon cannot be described by a Killing horizon. In the case when the space–time admits a timelike conformal Killing field, which becomes null on a boundary called the conformal stationary limit surface, one can locally describe the expanding event horizon by using this boundary, provided that it is a null geodesic hypersurface. In this case the boundary is called a conformal Killing horizon and is shown to be null and geodesic if and only if the twist of the conformal Killing trajectories on the hypersurface vanishes. Moreover if the space–time is conformally related to a stationary asymptotically flat black hole space–time, it is shown that this hypersurface is globally equivalent to the event horizon, provided that the conformal factor goes to a constant at null infinity. When the conformal stationary limit surface does not coincide with the conformal Killing horizon, a generalization of the weak rigidity theorem which establishes the conformal Killing property of the event horizon and the rigidity of its rotation is obtained. A physical definition of surface gravity for conformal Killing horizons is given, which is then used to formulate a generalized zeroth law of black hole physics.

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n−2)-mean curvature integral of its boundary. This question was considered by Santalo in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.

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 the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion).

Local rigidity of hyperbolic 3-manifolds after Dehn surgery

It is well known that some lattices in ${\rm SO}(n,1)$ can be nontrivially deformed when included in ${\rm SO}(n+1,1)$ (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in ${\rm SO}(3,1)$ which are locally rigid in ${\rm SO}(4,1)$ by considering closed hyperbolic $3$-manifolds obtained by Dehn filling on hyperbolic two-bridge knots. We generalize this result to Dehn filling on a more general class of one-cusped finite volume hyperbolic $3$-manifolds, allowing us to produce the first examples of closed hyperbolic $3$-manifolds which contain embedded quasi-Fuchsian surfaces but are locally rigid in ${\rm SO}(4,1)$.

Existence of prescribed mean curvature graphs in hyperbolic space

In this paper we are concerned with questions of existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space ?n+1. In the half-space setting, we will study radial graphs over the totally geodesic hypersurface \(\). We prove the following existence result: Let \(\) be a bounded domain of class \(\) and let \(\). If \(\) everywhere on \(\), where \(\) denotes the hyperbolic mean curvature of the cylinder over \(\), then for every \(\) there is a unique graph over \(\) with mean curvature \(\) attaining the boundary values \(\) on \(\). Further we show the existence of smooth boundary data such that no solution exists in case of \(\) for some \(\) under the assumption that \(\) has a sign.

Minimal submanifolds in Riemannian spin manifolds with parallel spinor fields

The aim of this paper is to investigate certain minimal submanifolds in a (2 n+ m)-dimensional Riemannian spin manifold with a nonzero parallel spinor field. In particular, we give a characterization of compact totally geodesic hypersurfaces in such an ambient space. We also study minimal surfaces in a four-dimensional hyperkähler manifold from the viewpoint of spinors. As a result, we recover two results about minimal tori in a four-dimensional flat torus and minimal surfaces in a four-dimensional nonflat hyperkähler manifold. A Lichnerowicz type formula on a submanifold plays a key role in this paper.

Variations d'entropies et déformations de structures conformes plates sur les variétés hyperboliques compactes

We study the link between variations of entropy on a compact hyperbolic manifold $M$ and infinitesimal flat conformal deformations of $M$. We remark that the entropy of the Liouville measure increases in the direction of these deformations. We then explain a new construction of infinitesimal flat conformal deformations by bending along totally geodesic hypersurfaces of $M$ which allows us to extend a theorem of L. Flaminio.

On Pappus-Type Theorems on the Volume in Space Forms

Let c be a curve in a n-dimensional space form Mλ n , let P t bea totally geodesic hypersurface of Mλ n orthogonal to c at c(t),and let D0 be a domain in P0. If D is thedomain in Mλ n obtained by a ‘motion’ of D0 alongc and D t is the domain in P t obtained by the motion ofD0 from 0 to t, we show that the n-volume ofD depends only on the length of the curve c, its first curvature, themodified (n − 1)-volume of D0 and the moment of D t with respect to the totally geodesic hypersurface of P t orthogonal to the normal vector f2(t) of c. As a consequence, if c(0) is the center of mass of D0, then the n-volume ofD is the product of the modified (n − 1)-volume of D0 and the length of c. We get an analogous theorem for ahypersurface of Mλ n obtained by ‘parallel motion’ of ahypersurface of P0 along c.

Singular operators satisfying an intertwining relation

An intertwining relation between the Beltrami-Laplace operator with an added potential and the Beltrami-Laplace operator is considered on a Riemannian manifold. It is shown that the potential singularities of codimension one form completely geodesic hypersurfaces.

Gravitational instantons of constant curvature

In this paper, we classify all closed flat 4-manifolds that have a reflective symmetry along a separating totally geodesic hypersurface. We also give examples of small-volume hyperbolic 4-manifolds that have a reflective symmetry along a separating totally geodesic hypersurface. Our examples are constructed by gluing together polytopes in hyperbolic 4-space.

Normal homogeneous spaces admitting totally geodesic hypersurfaces

Normal homogeneous spaces admitting totally geodesic hypersurfaces

Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms

In the last years, several authors have studied spacelike hypersurfaces with constant mean curvature in Lorentzian spaces of constant curvature, see for instance [2],[10], [11]. When the codimension of the spacelike submanifold is greater that one, the natural generalization, that is, the case of parallel mean curvature vector in the normal bundle, has been dealt in [1],[3],[5]. From a technical point of view, the closest case to that of spacelike hypersurfaces is when the codimension is equal to the index of the ambient space, so that, when the normal bundle is negative definite[1],[5],[9].Under this assumption, it has been mainly used as a tool classicalSimons' formula for the Laplacian of the length of the second fundamental form. However, this technique does not seen to be useful when the normal bundle is not definite.In [3],a different method has been introduced to study compact spacelike submanifolds with parallel mean curvature vector in de Sitter spaces, with non-definite normal bundle. As any index for the normal bundle was allowed, an assumption on the Ricci curvature (automatically satisfiedin the definite case) was shown to be necessary. In this paper we will study compact spacelike submanifolds with (nonzero) parallel mean curvature vector in a pseudo-Riemannian space form with Lorentzian normal bundle (of signature {l,p)). We use the same approach as in [3]; however no assumption on the curvature of the submaifold is now made, and the family of ambient spaces is extended in order to consider flat and negatively curved pseudo-Riemannian space forms. Our study was firstmotivated by the following easy fact: consider a totally umbilical and non-totally geodesic hypersurface Mn of a Riemannian space form of sectional curvature c e R, Nn+l(c). Embedding Nn+1(c) as a totally geodesic submanifold in an

Totally geodesic hypersurfaces of naturally reductive homogeneous spaces

Totally geodesic submanifolds of Riemannian symmetric spaces have been well investigated and it has been shown that they have beautiful and fruitful properties. In particular, due to the (M+,M_)-theory by B.Y. Chen and T. Nagano [1] this subject has made great progress. Naturally reductive homogeneous spaces are known as a natural generalization of Riemannian symmetric spaces. K. Tojo [6] investigated totally geodesic submanifolds of naturally reductive homogeneous spaces and obtained a necessary and sufficient condition of their existence. We will recall his result in section 3. Moreover he implicitly made the following conjecture.

Totally geodesic hypersurfaces in manifolds of nonpositive curvature

In this paper we determine the structure of an embedded totally geodesic hypersurfaceF or, more generally, of a totally geodesic hypersurfaceF without selfintersections under arbitrarily small angles in a compact manifoldM of nonpositive sectional curvature. Roughly speaking, in the case of locally irreducibleM the result says thatF has only finitely many ends, and each end splits isometrically asK×(0, ∞), whereK is compact.

Submanifolds with flat normal bundle in spaces of constant curvature

In the present paper parallel submanifolds and focal points of a given submanifold with flat normal bundle are discussed provided that the ambient space has constant sectional curvature. We present shape operators of parallel submanifolds with respect to arbitrary normal vectors. Furthermore, we prove that the focal points of a submanifold with flat normal bundle form totally geodesic hypersurfaces in the normal submanifolds.

Errata to "Hypersurfaces with Constant Mean Curvature in the Complex Hyperbolic Space"

Unfortunately, there is a mistake in the proof of Theorem 3.3 of our paper entitled Hypersurfaces with constant mean curvature in the complex space which appeared in the Transactions of the AMS (Vol. 339 (1993), 685-702). In the proof of this theorem we applied Hopf's maximum principle, which holds for a class of hypersurfaces satisfying an elliptic PDE, to a one-parameter family of hypersurfaces with constant mean curvature (cmc) of Q5 (using the notations of the paper) obtained by reflecting an initial cmc hypersurface of Q5 on a one-parameter family of totally geodesic hypersurfaces of Q5. However, while we know that the initial hypersurface satisfies an elliptic PDE since it is invariant by the S' group of isometries of Q5 and has cmc, the reflections of the hypersurface do not satisfy the equation since, although they have cmc, they are not S1 invariant any more. Therefore, Hopf's maximum principle cannot be used and the proof, as it stands in the paper, is not correct. This mistake was pointed out to us by Professor J. Eschenburg. So far, we have not found a way to correct it and, in fact, this seems to be a difficult question.

Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds

We show that a closed embedded totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. In particular, we show that ifM is a hyperbolic 3-manifold containingn rank two cusps andk disjoint totally geodesic embedded closed surfaces, then the volume ofM is bigger than $$(\tfrac{{\sqrt 3 }}{4})n + (4.4)k$$ . We also derive a (hyperbolic) quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last result, we construct examples to illustrate the qualitative sharpness of our tubular neighborhood function in dimension three. As an application of our results we give an eigenvalue estimate.

Particle Production by Tidal Forces and the Trace Anomaly

The quantum production of spinless particles 〈 n k ( t)〉, and of energy-momentum-stress 〈 T â b̂ ( P)〉 by the tidal forces of classical curved space-time are computed for the test case of 1 + 1 dimensions. Our new tools are gauge invariant normal ordering with respect to the energy measurable on a geodesic hypersurface, gauge invariant concepts of particle, vacuum, and of Fock space. Our computations are finite step by step, the predicted evolution of the energy-momentum tensor 〈 T â b̂ 〉 and the spectral energy density e k 〈 n k 〉 are consistent with each other throughout curved space-time, 〈 T â b̂ 〉 is covariantly conserved and has the standard trace anomaly R/24π for massless particles. The two chiralities, right-goers versus left-goers, are decoupled, the total 〈 T â b̂ 〉 is the sum of the chiral parts. An additional new tool is our "improved point-splitting method" which is equivalent to our gauge invariant normal ordering. We apply our methods to four problems: (1) The Rindler problem: we show that an accelerated observer in Minkowski space (without a detector in the Lagrangian) computes for the Minkowski vacuum 〈 T â b̂ 〉 = 0 and 〈 n k 〉 = 0. (2) An inhomogeneous patch of curvature (in an otherwise flat space-time) produces a burst of energy-momentum and of particles. This contradicts Parker′s result that there is no particle production if space-time is conformally flat and quantum fields are conformally coupled. (3) We compute the quantum production of energy density and pressure for a quantum field in external Friedmann-Robertson-Walker space-times in 1 + 1 dimensions. (4) We consider the gravitational field of a collapsing shell of classical matter in 3 + 1 dimensions, and we compute the production of Hawking radiation everywhere inside a linear wave guide in the radial direction. The results show that the horizon is not relevant for the generation of radiation and that the radiation is in a pure state on all spacelike hypersurfaces intersecting the shell before it crosses the gravitational radius.

The Orthogonal Spectrum of a Hyperbolic Manifold

Introduction. In this paper, we investigate the geometry of embedded hypersurfaces in hyperbolic manifolds of dimension greater than one. Our focus is on hypersurfaces that are either totally geodesic, horospherical (quotients of horospheres), or the boundary of a hyperbolic ball. These hypersurfaces have a natural decomposition into two pieces; one of which corresponds to base points of normal rays that satisfy a specified crossing condition, and the other piece being a disjoint union of embedded discs. We show that these embedded discs are in one to one correspondence with a certain set of lengths of orthogonals (the orthogonal spectrum) emanating from the hypersurface. In fact, the area of each disc is a monotonic function of the corresponding length in the spectrum. Thus the area (induced volume) of the hypersurface is intimitely related to the orthogonal spectrum (see theorem 1.1 for a precise statement). In special cases, (corollary 1.2) the piece of the hypersurface corresponding to the base points of normal rays is related to the limit set of the discrete isometry group that represents the manifold. As a consequence, we show that the measure of the limit set of this group is zero if and only if the area of the piece consisting of base points of normal rays is zero. The next theorem (theorem 1.3) relates the orthogonal spectrum of a closed embedded totally geodesic hypersurface with the lengths of geodesic loops based at some point of the ambient manifold. In particular, by applying a result of Sullivan, we can show that the geodesic flow acts ergodically on the unit tangent bundle of the manifold if and only if the sum of the areas of the discs embedded in the hypersurface diverges (corollary 1.4).