Existence and uniqueness of the solution to the Lp Minkowski problem for the electrostatic p-capacity are proved when p > 1 and 1 < p < n. These results are nonlinear extensions of the very recent solution to the Lp Minkowski problem for p-capacity when p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and the classical solution to the Minkowski problem for electrostatic capacity when p = 1 and p = 2 by Jerison.

The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et al.

We prove a sharp inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general $n \ge  3$. This inequality generalizes the classical Minkowski inequality [19] for surfaces in the three dimensional Euclidean space. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established in [4].

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.

We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvaturelike quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.