Abstract. We develop foundational theory for the Laplacian flow for closed G 2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound onwill imply bounds on all covariant derivatives of Rm and T . (2). We show that Λ(x, t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x, t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

Abstract. In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

Abstract. In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface Σ is strictly mean convex and star-shaped, then the flow hypersurface Σt converges to a large coordinate sphere as t → ∞ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structureφ on a compact 7-manifold M , the Laplacian flow with initial value in [φ], sufficiently close toφ, will converge to a point in the Diff 0 (M )-orbit ofφ. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.2010 Mathematics Subject Classification. 53C44, 53C25, 53C10.

In this paper, we first prove a compactness theorem for the space of closed embedded f -minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the f -Laplacian on compact manifold with positive m-Bakry-Émery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the n-shpere, or the n-dimensional hemisphere. Finally, for compact manifold with positive m-Bakry-Émery Ricci curvature and f -mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.2010 Mathematics Subject Classification. 53C42, 53C21.