265 research outputs found
Ancient solutions to the Ricci flow with pinched curvature
We show that any ancient solution to the Ricci flow which satisfies a
suitable curvature pinching condition must have constant sectional curvature.Comment: to appear in Duke Math Journa
Evolution equations of curvature tensors along the hyperbolic geometric flow
We consider the hyperbolic geometric flow introduced by Kong and Liu [KL]. When the Riemannian
metric evolve, then so does its curvature. Using the techniques and ideas of
S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection
and the curvature tensors along the hyperbolic geometric flow. The method and
results are computed and written in global tensor form, different from the
local normal coordinate method in [DKL1]. In addition, we further show that any
solution to the hyperbolic geometric flow that develops a singularity in finite
time has unbounded Ricci curvature.Comment: 15 page
Manifolds with 1/4-pinched flag curvature
We say that a nonnegatively curved manifold has quarter pinched flag
curvature if for any two planes which intersect in a line the ratio of their
sectional curvature is bounded above by 4. We show that these manifolds have
nonnegative complex sectional curvature. By combining with a theorem of Brendle
and Schoen it follows that any positively curved manifold with strictly quarter
pinched flag curvature must be a space form. This in turn generalizes a result
of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain
results for the case that the flag curvature is pinched with some constant
below one quarter, one of which generalizes a recent work of Petersen and Tao
Deformations of the hemisphere that increase scalar curvature
Consider a compact Riemannian manifold M of dimension n whose boundary
\partial M is totally geodesic and is isometric to the standard sphere S^{n-1}.
A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at
least n(n-1), then M is isometric to the hemisphere S_+^n equipped with its
standard metric. This conjecture is inspired by the positive mass theorem in
general relativity, and has been verified in many special cases. In this paper,
we construct counterexamples to Min-Oo's conjecture in dimension n \geq 3.Comment: Revised version, to appear in Invent. Mat
A compactness theorem for scalar-flat metrics on manifolds with boundary
Let (M,g) be a compact Riemannian manifold with boundary. This paper is
concerned with the set of scalar-flat metrics which are in the conformal class
of g and have the boundary as a constant mean curvature hypersurface. We prove
that this set is compact for dimensions greater than or equal to 7 under the
generic condition that the trace-free 2nd fundamental form of the boundary is
nonzero everywhere.Comment: 49 pages. Final version, to appear in Calc. Var. Partial Differential
Equation
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1
We prove that the hyperelliptic Torelli group is generated by Dehn twists about
separating curves that are preserved by the hyperelliptic involution. This verifies a
conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel
of the Burau representation evaluated at t = −1 and also the fundamental group of
the branch locus of the period mapping, and so we obtain analogous generating sets
for those. One application is that each component in Torelli space of the locus of
hyperelliptic curves becomes simply connected when curves of compact type are added
Pennsylvania Folklife Vol. 11, No. 1
• A Dunker Weekend Love Feast of 100 Years Ago • The Peacock in Pennsylvania • The Get-Togethers of the Young Amish Folk • Church and Meetinghouse Stables and Sheds • Abraham Harley Cassel - Dunkard Bibliophile • Mennonite Folklore • Springs and Springhouses • Finishing Wooden Surfaces • Early Funeral Notices • Collecting Dialect Folk Songshttps://digitalcommons.ursinus.edu/pafolklifemag/1006/thumbnail.jp
- …