Dynamical stability and instability of Ricci-flat metrics

In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer, getting rid of the integrability assumption.Comment: 6 page

A compactness theorem for complete Ricci shrinkers

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.Comment: 28 pages, final version, to appear in GAF

The stability inequality for Ricci-flat cones

In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cones over products of Einstein manifolds and over Kähler-Einstein manifolds with h^{1,1}>1 are unstable in dimension less than 10. As results of independent interest, our computations indicate that the Page metric and the Chen-LeBrun-Weber metric are unstable Ricci shrinkers. As a final bonus, we give plenty of motivations, and partly confirm a conjecture of Tom Ilmanen relating the lambda-functional, the positive mass theorem and the nonuniqueness of Ricci flow with conical initial data

A Perspective on Resource Synchronization

Web applications frequently leverage resources made available by remote web servers. As resources are created, updated, deleted, or moved, these applications face challenges to remain in lockstep with changes on the server. Several approaches exist to help meet this challenge for use cases where good enough synchronization is acceptable. But when strict resource coverage or low synchronization latency is required, commonly accepted Web-based solutions remain illusive. This paper provides a perspective on the resource synchronization problem that results from inspiration gained from prior work, and initial insights resulting from the recently launched NISO/OAI ResourceSync effort

A Technical Framework for Resource Synchronization

This is the second paper in D-Lib Magazine about the ResourceSync effort conducted by the National Information Standards Organization (NISO) and the Open Archives Initiative (OAI). The first part provided a perspective on the resource synchronization problem and introduced a template that organized possible components of a resource synchronization framework in a modular manner. This paper details a technical framework devised using that template

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let $M$ be a complete Riemannian manifold possibly with a boundary \pp M. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:=\DD+Z, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of \pp M if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm \|\Ric_Z\|_\infty on manifolds without boundary. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first named author, such that the proofs are significantly simplified