Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Functions of one complex variable II

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شیوع عوارض حاد تزریق خون در بیماران بستری در 11 بیمارستان تهران و مازندران

This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian submanifolds. The examples are constructed using a special class of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer homology. 
Revised version: one footnote removed, one reference added

Lagrangian two-spheres can be symplectically knotted

We develop foundational theory for the Laplacian flow for closed G2 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

 $$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$
 will imply bounds on all covariant derivatives of Rm and T. (2). We show that $${\Lambda(x,t)}$$
 will blow up at a finite-time singularity, so the flow will exist as long as $${\Lambda(x,t)}$$
 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) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

/pdf/laplacian-flow-for-closed-g2-structures-shi-type-estimates-163bue10z8.pdf

Laplacian flow for closed G2 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 $\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ 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.

/pdf/laplacian-flow-for-closed-g-2-structures-shi-type-estimates-2lbvsemkux.pdf

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

We prove that torsion-free G2 structures are (weakly) dy- namically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structure ¯ ' on a compact 7-manifold M, the Laplacian flow with initial value in (¯ '), sufficiently close to ¯ ', will converge to a point in the Diff 0 (M)-orbit of ¯ '.

Stability of torsion-free g2 structures along the laplacian flow

Let $\varphi(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed positive time $t\in (0,T]$, $(M,\varphi(t),g(t))$ is real analytic, where $g(t)$ is the metric induced by $\varphi(t)$. Consequently, any Laplacian soliton is real analytic and we obtain unique continuation results for the flow.

Laplacian flow for closed G_2 structures: real analyticity

This article studies the deformation theory of coassociative 4-folds N with conical singularites in a G(2) manifold. We describe three moduli spaces: first we consider deformations with the same singularities as N, then allow for changes in the singularities and, finally, include variations of the ambient G2 structure. We show that the moduli space, in each case, is locally homeomorphic to the kernel of a smooth map between smooth manifolds and determine a lower bound for its expected dimension. Further, by relaxing the condition on the G2 structure, we prove a generic smoothness result for the second and third moduli spaces.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2007/0015/0005/CAG-2007-0015-0005-a001.pdf

Jason D. Lotay

Papers

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

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

Stability of torsion-free g2 structures along the laplacian flow

Laplacian flow for closed G_2 structures: real analyticity

Coassociative 4-folds with conical singularities