15,533 research outputs found

    Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

    Get PDF
    We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.Comment: Typos corrected, some improved bound

    Nonlinear geometric analysis on Finsler manifolds

    Full text link
    This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the Γ\Gamma-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.Comment: 37 pages, to appear in a topical issue of European Journal of Mathematics "Finsler Geometry: New Methods and Perspectives". arXiv admin note: text overlap with arXiv:1602.0039

    Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

    Get PDF
    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 on Λ(x,t)=(∣∇T(x,t)∣g(t)2+∣Rm(x,t)∣g(t)2)12\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12} will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t)\Lambda(x,t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t)\Lambda(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). (5). Finally, we study compact soliton solutions of the Laplacian flow.Comment: 59 pages, v2: minor corrections and additions, accepted version for GAF
    • …
    corecore