Let be an elliptically fibered K3 surface, admitting a sequence of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to . Given the corresponding sequence of Hermitian-Yang-Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence converges to a flat connection . Furthermore, if the restriction is of the form for n distinct points , then these points uniquely determine .
In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is -rectifiable, for , if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of -rectifiable measures. Namely, we prove that the support of a -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of -rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup.
We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in , which include Lipschitz submanifolds.
We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the -doubly warped products of smooth measure spaces and establish -Bakry-Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：