Abstract. In this paper we study nonparametric mean curvature type flows in M×R which are represented as graphs (x, u(x, t)) over a domain in a Riemannian manifold M with prescribed contact angle. The speed of u is the mean curvature speed minus an admissible function ψ(x, u, Du).Long time existence and uniformly convergence are established if ψ(x, u, Du) ≡ 0 with vertical contact angle and ψ(x, u, Du) = h(x, u)ω with hu(x, u) ≥ h0 > 0 and ω = 1 + |Du| 2 . Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in M 2 which is a surface with nonnegative Ricci curvature.

Abstract. We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor φ(r). If φ ′ (r) > 0 and φ ′′ (r) ≥ 0, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of φ ′′ (r) and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.

In this note we discuss graphs over a domain Ω ⊂ N 2 in the product manifold N 2 ×R. Here N 2 is a complete Riemannian surface and Ω has piece-wise smooth boundary. Let γ ⊂ ∂Ω be a smooth connected arc and Σ be a complete graph in N 2 ×R over Ω. We show that if Σ is a minimal or translating graph, then γ is a geodesic in N 2 . Moreover if Σ is a CMC graph, then γ has constant principle curvature in N 2 . This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and CMC graphs in N 2 ×R.

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We study curve shortening flows in two types of warped product manifolds. These manifolds are S 1 ×N with two types of warped metrics where S 1 is the unit circle in R 2 and N is a closed Riemannian manifold. If the initial curve is a graph over S 1 , then its curve shortening flow exists for all times and finally converges to a geodesic closed curve.