We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de~Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de~Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect \(n|h|^2-H^2\) with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.
We study nonlinear mean-curvature spectral equations for hypersurfaces in Euclidean, spherical and hyperbolic space. For an admissible analytic response $F$ and ambient sectional curvature $c$, we prove minimality when the spectral parameter satisfies $\sigma\ge nc$, and strict mean-curvature bounds at gradient points...
This paper studies self-shrinkers and the long-time behavior of the inverse $\sigma_k$ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of se...
We establish the free-boundary version of Schoen-Simon's regularity and compactness theory for stable minimal hypersurfaces in a complete $(n+1)$-dimensional Riemannian manifold-with-boundary, following the intrinsic PDE scheme introduced by the first-named author. The key step is to prove sheeting theorems in the two...
C. Bellettini, M. Li, Davide Parise et al.· 0 citations
We prove that a smooth, regular, asymptotically flat stationary vacuum black hole has a nonextremal Kerr exterior if no null geodesic orthogonal to its stationary Killing field is trapped, establishing the rigidity conjecture of Alexakis, Ionescu, and Klainerman. In the rotating case, we start with the rotational Killi...
We study static, spherically symmetric traversable wormholes in trace-coupled gravity with action density f(R,T) = R + lambda T^2. The spacetime is built from C-infinity deformations of Schwarzschild that vanish identically outside a finite core, so the exterior is exactly Schwarzschild and no thin shell or junction su...
Mushtaq Ahmad, M. Shamir, A. Althukair· 0 citations
We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz--Minkowski three-space. On the hyperbolic plane, solutions of $\Delta u-2u=0$ give rise to branched spacelike maximal surfaces. We prove that, on a bounded...
M'arcio Batista, Iury Domingos· 0 citations
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