Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn exotic sphere with positive curvature
245
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive sectional curvature on a homotopy (but not diffeomorphic) $RP^7$.
rate research
Read More
In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $mathbb{S}^4,$ or $mathbb{R}mathbb{P}^4$ or quotients of $mathbb{S}^3times mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $mathbb{S}^3times mathbb{R}$, or a connected sum of them.
Gromoll and Meyer have represented a certain exotic 7-sphere $Sigma^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Sigma^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $geq 1$ which we describe explicitly.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Bohm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of Brendle-Schoens recent differential sphere theorem, bypassing any discussion of positive isotropic curvature.
Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,alpha}$ convergence techniques, we prove several differentiable stability and sphere theor
We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomolnyi equations, corresponding to $c=0$, in all admissible Kahler classes. Our second main result completely solves the existence problem for $c>0$. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with $c = 0$ and deforms the Kahler class. For the latter result we start from the established solution in any fixed admissible Kahler class and deform the coupling constant $alpha$ towards $0$. A salient feature of our argument is a new bound $S_g geqslant c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
