Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-withboundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclid...

An increased role for rail freight is an objective of the British government. Limited growth potential exists in rail’s traditional bulk markets, so more non-bulk volume is needed. This paper focuses on non-bulk rail freight activity in Britain, through deskbased research and company interviews. It considers changes in both the intermodal and traditional less-than-trainload (LTL) markets over t...

Let F : Σ n × [0, T) → R n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m)...

of ann-dimensional manifold without boundary into Euclidean space. We say Af = x(Mn) moves by mean curvature if there exists a one-parameter family Xt = x( ·, t) of immersions with corresponding images J..!I1 = x 1(lvfn) satisfying d dtx(p,t) = -H(p,t)v(p,t) (1) x(p,O) = xo(P) for some initial data xo. Here H(p,t) and v(p,t) denote mean curvature and outer unit normal of the hypersurface M 1 at...

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on that of [Th]. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of...