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In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$., To appear in Rev. Mat. Iberoam

A translation operator acting in a space with a diagonal metric is introduced to describe the motion of a particle in a quantum system. We show that the momentum operator and, as a consequence, the uncertainty relation now depend on the metric. It is also shown that, for any metric expanded up to second order, this formalism naturally leads to an extended uncertainty principle (EUP) with a minimum momentum dispersion. The Ehrenfest theorem is modified to include an additional term related to a tidal force arriving from the space curvature introduced by the metric. For one-dimensional systems, we show how to map a harmonic potential to an effective potential in Euclidean space using different metrics.

The reduced thin-sandwich equations (RTSE) appear within Wheeler’s thin-sandwich approach toward the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed, in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighborhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper, we will analyze such a set of equations on the manifolds Euclidean at infinity in a neighborhood of asymptotically Euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parameterize an open subset in the space of solutions of the ECE. In addition, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.

We introduce an extended notion of mean curvature for graphs via the Euler-Lagrange equation of a geometric elliptic functional. We then draw some geometric conclusions for Killing graphs with prescribed weighted and anisotropic mean curvatures with the aid of a general form of the weak maximum principle and a sufficient condition for an appropriate notion of parabolicity.

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in two ways. On the one hand, we show that the results obtained by the mentioned authors are valid in arbitrary dimensions, and on the other hand we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n greater or equal to 3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed., Comment: 17 pages

We prove the existence of classical solutions to the Dirichlet prob- lem for a class of fully nonlinear elliptic equations of curvature type on Riemann- ian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck (4) and Ivochkina, Trudinger and Lin (18), (19), (25) to more general curva- ture functions under mild conditions on the geometry of the domain.

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ℝn+1 are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly enough at infinity. Combining this with a computation of the (r+1)-curvature of normal graphs and the deformation theory in weighted Holder spaces developed by Mazzeo, Pacard, Pollack, Uhlenbeck and others, we produce new infinite dimensional families of r-minimal hypersurfaces in ℝn+1 obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space \({\mathcal{M}}_{r,k}(g)\) of elliptic r-minimal hypersurfaces with k≥2 ends of planar type in ℝn+1 endowed with an ALE metric g, and show that \({\mathcal{M}}_{r,k}(g)\) is an analytic manifold of formal dimension k(n+1), with \({\mathcal{M}}_{r,k}(g)\) being smooth for a generic g in a neighborhood of the standard Euclidean metric.

In this paper we will discuss a Weierstrass type representation for minimal surfaces in Riemannian and Lorentzian 3-dimensional manifolds.

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups.

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU 1 , 1 endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R 2 , 2 . We prove that the Gauss map of a minimal surface immersed in SU 1 , 1 is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.

We prove a Bonnet theorem for isometric immersions of semi-Riemannian manifolds into products of semi-Riemannian space forms. Namely, we give necessary and sufficient conditions for the existence and uniqueness (up to an isometry of the ambient space) of an isometric immersion of a semi-Riemannian manifold into a product of semi-Riemannian space forms.

It is proved that a pair of spinors satisfying a Dirac type equation represents surfaces immersed in Berger spheres with prescribed mean curvature. Using this, we prove that the Gauss map of a minimal surface immersed in a Berger sphere is harmonic. Conversely, we exhibit a representation of minimal surfaces in Berger spheres in terms of a given harmonic map. The examples we constructed appear in associated families.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

Given a compact Riemannian manifold M, we consider a warped product M = I × h M where I is an open interval in ℝ. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function ψ in M, we find a closed hypersurface ∑ which is solution of an equation of the form F(B) = ψ where B is the second fundamental form of ∑ and F is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature.

The subject of this paper is properly embedded H − H- surfaces in Riemannian three manifolds of the form M 2 × R M^2\times \mathbf {R} , where M 2 M^2 is a complete Riemannian surface. When M 2 = R 2 M^2={\mathbf R}^2 , we are in the classical domain of H − H- surfaces in R 3 {\mathbf R}^3 . In general, we will make some assumptions about M 2 M^2 in order to prove stronger results, or to show the effects of curvature bounds in M 2 M^2 on the behavior of H − H- surfaces in M 2 × R M^2\times \mathbf {R} .

It is proved that an embedded hypersurface in a hemisphere of the Euclidean unit spherewith constant mean curvature and spherical boundary inherits, under certainconditions, the symmetries of its boundary. In particular, spherical caps are theonly such hypersurfaces whose boundary are geodesic spheres.

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere \(\mathbb{S}^{n + 1} (a)\) of radius a. Given a compact domain Ω, with some conditions, contained in a totally geodesic sphere S of \(\mathbb{S}^{n + 1} (a)\) and a real differentiable function χ on Ω, we define the graph of χ in \(\mathbb{S}^{n + 1} (a)\) considering the ‘height’ χ(x) on the minimizing geodesic joining the point x of Ω to a fixed pole of \(\mathbb{S}^{n + 1} (a)\). For a real number H such that |H| is bounded for a constant depending on the mean curvature of the boundary Γ of Ω and on a fixed number δ in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary Γ, whenever the diameter of Ω is smaller than a constant depending on δ. If we have conditions on Γ, that is, let Γ′ be a graph over Γ of a function, if |H| is bounded for a constant depending only on the mean curvature of Γ and if the diameter of Ω is smaller than a constant depending on H and Γ, then we prove that there exists a unique graphs with mean curvature H and boundary Γ′. The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space Hn+1 defined over domains in geodesic spheres of Hn+1 whose boundary has positive mean curvature with respect to the inward orientation.

We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for modelling stationary hypersurfaces under the influence of a non-homogeneous gravitational field defined over an arbitrary Riemannian manifold.

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.

We use geometric analysis of the Lorentzian distance on an n-dimensional spacetime to derive some non-existence results and some sharp mean curvature estimates for trapped submanifolds contained in the chronological future of either a point or of a space-like achronal hypersurface . Our results are given as an application of the corresponding Hessian comparison results for the Lorentzian distance function and the so-called weak maximum principle for the Laplacian operator.

We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a trivial minimal graph which contacts the cylinder over the domain orthogonally along its boundary.

It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products $\mathbb{S}^2\times\Rr$ and $\mathbb{H}^2\times \Rr$ discovered by U. Abresch and H. Rosenberg could be obtained as a linear combination of usual Hopf differentials. Using this fact, we are able to extend it for Lorentzian products. Families of examples of helicoidal CMC surfaces on these spaces are explicitly described. We also present some characterizations of CMC rotationally invariant discs and spheres. Finally, after establish some height and area estimates, we prove the existence of constant mean curvature Killing graphs., 38 pages

Let $\psi$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $\psi$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980's. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigate the problem of uniqueness in hyperbolic space and show that uniqueness (up to a geometrically trivial transformation) holds under the same conditions under which existence was established., Comment: 12 pages

We obtain a priori bounds for solutions of the nonlinear second-order elliptic equation of the geometric problem consisting in finding a compact starshaped hypersurface in a space form whose mth elementary symmetric function of principal curvatures is a given function.

We establish a spinorial representation for surfaces immersed with prescribed mean curvature in Heisenberg space. This permits to obtain minimal immersions starting with a harmonic Gauss map whose target is either the Poincare disc or a hemisphere of the round sphere.