Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p>n$ are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data $\sigma$ and $\tau$. We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on $1$-forms. This last equation can be shown to be without solutions in many situations. As a corollary, we get the existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which, in particular, hold on a dense set of metrics $g$ for the $C^0$-topology.

To every matroid, we associate a class in the $K$-theory of the Grassmannian. We study this class by using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend the second author’s results concerning the behavior of such classes under direct sum, series and parallel connection, and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a three-sphere which has scalar curvature greater than or equal to $6$ and is not round must have an embedded minimal sphere of area strictly smaller than $4\pi$ and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

Let $L\to X$ be an ample bundle over a compact complex manifold. Fix a Hermitian metric in $L$ whose curvature defines a Kähler metric on $X$. The Hessian of Mabuchi energy is a fourth-order elliptic operator ${\mathcal{D}}^{\ast }\mathcal{D}$ on functions which arises in the study of scalar curvature. We quantize ${\mathcal{D}}^{\ast }\mathcal{D}$ by the Hessian $P_k^{\ast }{P}_k$ of balancing energy, a function appearing in the study of balanced embeddings. $P_k^{\ast }{P}_k$ is defined on the space of Hermitian endomorphisms of $H^0(X,L^k)$ endowed with the $L^2$-inner product. We first prove that the leading order term in the asymptotic expansion of $P_k^{\ast }{P}_k$ is ${\mathcal{D}}^{\ast }\mathcal{D}$. We next show that if $Aut(X,L)/{\mathbb{C}}^{\ast }$ is discrete, then the eigenvalues and eigenspaces of $P_k^{\ast }{P}_k$ converge to those of ${\mathcal{D}}^{\ast }\mathcal{D}$. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kähler metric. As consequences of our results we prove that an estimate of Phong and Sturm is sharp and give a negative answer to a question posed by Donaldson. We also discuss some possible applications to the study of Calabi flow.