A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.
We prove a $C^0$ a priori estimate on a solution of the quaternionic Calabi problem on an arbitrary compact connected HKT-manifold. This generalizes earlier works where this result was proven under certain extra assumptions on the manifold.
The goal of this paper is to describe the $alpha$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $alpha$ the $alpha$-cosine transform is a composition of the $(alpha+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $alpha$ except one we interpret the $alpha$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $alpha$, which is $-(min{i,n-i}+1)$, is still an open problem.
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
There is a well known construction of weakly continuous valuations on convex compact polytopes in R^n. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of R^n. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R^3 more explicit results are obtained.
A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion). In this paper this problem is solved under an extra assumption that the manifold admits a flat hyperKaehler metric compactible with the underlying hypercomplex structure. The proof uses the continuity method and a priori estimates.
Recently an algebra of smooth valuations was attached to any smooth manifold. Roughly put, a smooth valuation is finitely additive measure on compact submanifolds with corners which satisfies some extra properties. In this note we initiate a study of modules over smooth valuations. More specifically we study finitely generated projective modules in analogy to the study of vector bundles on a manifold. In particular it is shown that on a compact manifold there exists a canonical isomorphism between the $K$-ring constructed out of finitely generated projective modules over valuations and the classical topological $K^0$-ring constructed out of vector bundles.
On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Amp`ere operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property.
