Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa) = \sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l = \sigma$; $1\leq l < n$ in both deSitter and Hyperbolic space.Comment: 24 page

Entire downward translating solitons to the mean curvature flow in Minkowski space

In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $\Sigma=\{(x, u(x))| x\in\mathbb{R}^n\}$ is a strictly spacelike hypersurface, then $\Sigma$ reduces to a strictly convex rank k soliton in $\mathbb{R}^{k, 1}$ (after splitting off trivial factors) whose "blowdown" converges to a multiple $\lambda\in(0, 1)$ of a positively homogeneous degree one convex function in $\mathbb{R}^k$. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation

Charged cosmological dust solutions of the coupled Einstein and Maxwell equations

It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupled Einstein and Maxwell equations admit a static multiple blackhole solution representing a balanced equilibrium state of finitely many point charges. This is a result of the exact cancellation of gravitational attraction and electric repulsion under an explicit condition on the mass and charge ratio. The resulting system of particles, known as an extremely charged dust, gives rise to examples of spacetimes with naked singularities. In this paper, we consider the continuous limit of the Majumdar--Papapetrou--Hartle--Hawking solution modeling a space occupied by an extended distribution of extremely charged dust. We show that for a given smooth distribution of matter of finite ADM mass there is a continuous family of smooth solutions realizing asymptotically flat space metrics

Hypersurfaces of constant curvature in Hyperbolic space

We show that for a very general and natural class of curvature functions, the problem of finding a complete strictly convex hypersurface satisfying f({\kappa}) = {\sigma} over (0,1) with a prescribed asymptotic boundary {\Gamma} at infinity has at least one solution which is a "vertical graph" over the interior (or the exterior) of {\Gamma}. There is uniqueness for a certain subclass of these curvature functions and as {\sigma} varies between 0 and 1, these hypersurfaces foliate the two components of the complement of the hyperbolic convex hull of {\Gamma}

A note on starshaped compact hypersurfaces with a prescribed scalar curvature in space forms

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $\sigma_2(\kappa(X))=f(X, \nu(X)).$ In this note, we give a simpler proof of this result, and extend it to space forms

Self-shrinkers to the mean curvature flow asymptotic to isoparametric cones

In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface $\Gamma \subset S^n$. The theory of isoparametic hypersurfaces is extremely rich and there are infinitely many distinct classes of examples, each with infinitely many members

Rearrangements and radial graphs of constant mean curvature in hyperbolic space

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques

Hypersurfaces of Constant Curvature in Hyperbolic Space I

We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions

Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $\Gamma$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $\Gamma$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate".Comment: 20 page

A priori estimates for semistable solutions of semilinear elliptic equations

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound follows from a $L^p$-estimate for $f'(u)$ for every $p<3$ and $n\geq 2$. Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.Comment: One bibliographical reference has been corrected with respect to the previous versio