We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Råde [21] and the bubbling criterion in dimension 4 of Struwe [23] in the case where the initial flow data is smooth.

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Following [3], we define a notion of entropy for connections over R n which has shrinking Yang-Mills solitons as critical points. As in [3], this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying "generic singularities" of Yang-Mills flow, and we discuss the differences in this strategy in dimension n = 4 versus n ≥ 5.

Abstract. Inspired by work of Colding-Minicozzi [3] on mean curvature flow, Zhang [16] introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable F-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang [11] to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.

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Abstract. We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the kth tensor powers of a positive line bundle L in a 1 √ kneighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential kϕ in a 1 √ k -neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.

Suppose 0 < p ≤ 2 and that (Ω, µ) is a measure space for which Lp(Ω, µ) is at least twodimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω, µ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω, µ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the formwhere {z 1 , . . . , zn} is a subset of Lp(Ω, µ) and α 1 , . . . , αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω, µ). The cases p < 2 and p = 2 are substantially different and are treated separately. The case p = 1 generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin.Another reason for studying non-trivial p-polygonal equalities in Lp(Ω, µ) is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if (X, d) is a metric space that has strict q-negative type for some q ≥ p, then: (1) (X, d) is not isometric to any linear subspace W of Lp(Ω, µ) that contains a pair of disjointly supported non-zero vectors, and (2) (X, d) is not isometric to any subset of Lp(Ω, µ) that has non-empty interior. Furthermore, in the case p = 2, it also follows that (X, d) is not isometric to any affinely dependent subset of L 2 (Ω, µ). More generally, we show that if (Y, ρ) is a metric space whose generalized roundness ℘ is finite and if (X, d) is a metric space that has strict q-negative type for some q ≥ ℘, then (X, d) is not isometric to any metric subspace of (Y, ρ) that admits a non-trivial p 1 -polygonal equality for some p 1 ∈ [℘, q]. It is notable in all of these statements that the metric space (X, d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin.We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.2010 Mathematics Subject Classification. 54E40, 46B04, 46C05.

We study singularity structure of Yang-Mills flow in dimensions n ≥ 4. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set.Date: February 1, 2016. 1 A corollary of these theorems is the existence of a either Yang-Mills connection or Yang-Mills soliton as a blowup limit of arbitrary finite time singularities. For type I singularities the existence of soliton blowup limits was established in [24], following from the entropy monotonicity for Yang-Mills flow demonstrated in [9]. The existence of soliton blowup limits for arbitrary singularities of mean curvature flow was established in [11], relying on the structure theory associated with Brakke's weak solutions. A preliminary investigation into the entropy-stability of Yang-Mills solitons was undertaken in [3] and [12]. Those results now apply to studying arbitrary finite-time singularities of Yang-Mills flow, as all admit singularity models which are either Yang-Mills connections or Yang-Mills solitons. Corollary 1.4. Fix n ≥ 4 and let E → (M n , g) be a vector bundle over a closed Riemannian manifold. Let ∇ t a smooth solution to Yang-Mills flow on [0, T ) such that lim sup t→T |F ∇tsuch that the corresponding blowup sequence converges modulo gauge transformations to either (1) A Yang-Mills connection on S 4 .(2) A Yang-Mills soliton.