In this paper we establish a volume comparison theorem for cocentric metric balls at arbitrary point for manifolds with asymptotically nonnegative Ricci curvature, which will allow us to prove the finiteness of the number of ends.

As an application, we compute the Eells–Kuiper and t-invariants of certain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomorphism type of a new manifold of positive sectional curvature.

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem...

In this note we analyze equilibria of static Stefan type problems with crystalline/singular weighted mean curvature in the plane. Our main goal is to improve the meaning of variational solutions so that their properties allow us to call them almost classical solutions. The idea of our approach is based on a new definition of a composition of multivalued functions.

Let $(M,g)$, $n\ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0\<l\le L$, we prove that there exists $\epsilon =\epsilon (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies$$l\le s\le L$$and the Einstein tensor satisfies$$\left|\mathrm{Ric}-\frac{s}{n}g\right|\le \epsilon$$then $M$ is diffeomorphic to a symmetric space of compact type.This is related to the result of S. Brendle on the metric rigidity of Einstein manifolds with nonnegative isotropic curvature.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1\left(L\right)$ and its automorphism group $\mathrm{Aut}(V,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where $\mathrm{Aut}(V,L)$ is not discrete.

Let $(M,g)$ be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar’s work. Moreover, we reduce the remaining cases to the positive mass theorem.

We consider in ${ℝ}^2$ all curvature equation $\frac{\partial u}{\partial t}=\left|Du\right|G\left(\mathrm{curv}\left(u\right)\right)$ where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respect to any contrast change u → g(u), with g nondecreasing. Consider the contrast invariant operator $T_t:u_o\to u\left(t\right)$. A Matheron theorem asserts that all contrast invariant operator T can be put in a form $\left(Tu\right)\left(𝐱\right)={inf}_{B\in ℬ}{sup}_{𝐲\in B}u(𝐱+𝐲)$. We show the asymptotic equivalence of both formulations. More precisely, we show that all curvature equations can be obtained...

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation ${\square }_{\mathbf{g}}\phi =0$, where $\gg$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor $\mathbf{R}$ of $\gg$ is a major step towards the proof of the bounded $L^2$ curvature conjecture.