We show that a properly immersed minimal hypersurface in M × R+ equals some M×{c} when M is a complete, recurrent ndimensional Riemannian manifold with bounded curvature. If on the other hand,M is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M.

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the quotient Finsler structure, which is dual in a sense to the inclusion of a convex surface in a normed space as a submanifold. It has an associated quotient girth, which is similar to the notion of girth defined by Sch¨affer. We prove the analogs of Sch¨affer’s dual girth conjecture (proved by ´ Alvarez-Paiva) and the Holmes–Thompson dual volumes theorem in the quotient setting. We then show that the quotient Finsler structure admits a natural extension to higher Grassmannians, and prove the corresponding theorems in the general case. We follow ´ Alvarez-Paiva’s approach to the problem, namely, we study the symplectic geometry of the associated co-ball bundles. For the higher Grassmannians, the theory of Hamiltonian actions is applied.

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in$Torchiani, C.$,$Line Bundles on Tropical Varieties$, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern,2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.

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 [8] and [11].

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa.