In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman’s ν-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.

We obtain an estimate for the volumes of neighborhoods of sets of large curvature in three-dimensional K¨ahler-Einstein manifolds. The key technical step is to prove that a version of monotonicity for L2 energy holds as long as the underlying region does not “carry homology” (in the sense that the normalized energy in a ball controls the normalized energy in an interior ball).

Let $ f : \sum_1 \to \sum_2$ be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in $\sum_1 \times \sum_2$2 by the mean curvature flow. Under suitable conditions on the curvature of $\sum_1$ and $\sum_2$ and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant $t$, the flow remains the graph of a map $f_t$ and $f_t$ converges to a constant map as $t$ approaches infinity. This also provides a regularity estimate for Lipschtz initial data.

We extend recent existence results for compact constant mean curvature normal geodesic graphs with boundary in space forms in a unified way to a large class of ambient spaces. That extension is achieved by solving a constant mean curvature existence problem in a general setting presented in two isometrically equivalent versions.

In this paper we construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of S2 × S3. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A K¨ahler–Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi–Boyer–Galicki method.