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Let be a compact Riemannian manifold on which a trace-free and divergence-free and a positive function , are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data and . We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on -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 for the -topology.
To every matroid, we associate a class in the -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 and is not round must have an embedded minimal sphere of area strictly smaller than and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.
Let be an ample bundle over a compact complex manifold. Fix a Hermitian metric in whose curvature defines a Kähler metric on . The Hessian of Mabuchi energy is a fourth-order elliptic operator on functions which arises in the study of scalar curvature. We quantize by the Hessian of balancing energy, a function appearing in the study of balanced embeddings. is defined on the space of Hermitian endomorphisms of endowed with the -inner product. We first prove that the leading order term in the asymptotic expansion of is . We next show that if is discrete, then the eigenvalues and eigenspaces of converge to those of . 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.