Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kahler. We prove that if |Rm(g(t))| is bounded by a/t for some a > 0, then g(t) is Kahler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kahler-Ricci flow on a non-compact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and |Rm(g(t))| ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to C^n; and (ii) there is ε(n) > 0 depending only on n such that if (M^n,g_0) is a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+ε(n))^{−1}h ≤ g_0 ≤ (1+ε(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C^n.

Multistep cell fate transitions with stepwise changes of transcriptional profiles are common to many developmental, regenerative and pathological processes. The multiple intermediate cell lineage states can serve as differentiation checkpoints or branching points for channeling cells to more than one lineages. However, mechanisms underlying these transitions remain elusive. Here, we explored gene regulatory circuits that can generate multiple intermediate cellular states with stepwise modulations of transcription factors. With unbiased searching in the network topology space, we found a motif family containing a large set of networks can give rise to four attractors with the stepwise regulations of transcription factors, which limit the reversibility of three consecutive steps of the lineage transition. We found that there is an enrichment of these motifs in a transcriptional network controlling the early T cell development, and a mathematical model based on this network recapitulates multistep transitions in the early T cell lineage commitment. By calculating the energy landscape and minimum action paths for the T cell model, we quantified the stochastic dynamics of the critical factors in response to the differentiation signal with fluctuations. These results are in good agreement with experimental observations and they suggest the stable characteristics of the intermediate states in the T cell differentiation. These dynamical features may help to direct the cells to correct lineages during development. Our findings provide general design principles for multistep cell linage transitions and new insights into the early T cell development. The network motifs containing a large family of topologies can be useful for analyzing diverse biological systems with multistep transitions.

Let X be a compact toric Kaehler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula which equates such open GW invariants with closed GW invariants of certain X-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of X, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.

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