It is shown that within the Lp-Brunn–Minkowski theory that
Aleksandrov’s integral curvature has a natural Lp extension, for all
real p. This raises the question of finding necessary and sufficient
conditions on a given measure in order for it to be the Lp-integral
curvature of a convex body. This problem is solved for positive p
and is answered for negative p provided the given measure is even.
In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least 1 co-dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone C––ε to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before C––ε, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of non-spherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.
We consider a characteristic problem of the vacuum Einstein
equations with part of the initial data given on a future asymptotically
flat null cone, and show that the solution exists uniformly
around the null cone for general such initial data. Therefore, the
solution contains a piece of the future null infinity. The initial
data are not required to be small and the decaying condition is
consistent with those in the works of  and .
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a ﬁnite dimensional variational principle.
Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on C3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity C2/Z2×C, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.