We extend Schmid’s $\rm{SL}_2$-orbit theorem to a class of variations of mixed Hodge structure which normal functions, logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as a consequence of this theorem, we obtain a simple formula for the asymptotic behavior of the archimedean height of a flat family of algebraic cycles which depends only on the weight filtration and local monodromy.

We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of $\mathbb{R}^3$ by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two-sided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds were first established by Hamilton. Later on, De Turck gave a simplified proof. In the later part of 80’s, Shi generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.

We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.