15,533 research outputs found
Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations
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
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 -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
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 will imply bounds on all
covariant derivatives of Rm and T. (2). We show that will blow
up at a finite-time singularity, so the flow will exist as long as
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
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