Given a Hermitian line bundle $$L\rightarrow M$$ L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$ ϵ → 0 , of couples $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ E ϵ ( u , ∇ ) = ∫ M ( | ∇ u | 2 + ϵ 2 | F ∇ | 2 + 1 4 ϵ 2 ( 1 - | u | 2 ) 2 ) of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$ ∇ is a Hermitian connection on L with curvature $$F_{\nabla }$$ F ∇ . Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$ lim sup ϵ → 0 E ϵ ( u ϵ , ∇ ϵ ) < ∞ , we show that the energy measures converge subsequentially to (the weight measure $$\mu $$ μ of) a stationary integral $$(n-2)$$ ( n - 2 ) -varifold. Also, we show that the $$(n-2)$$ ( n - 2 ) -currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$ 2 π Γ , for an integral $$(n-2)$$ ( n - 2 ) -cycle $$\Gamma $$ Γ with $$|\Gamma |\le \mu $$ | Γ | ≤ μ . Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$ ( n - 2 ) -varifolds in an arbitrary closed Riemannian manifold.

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.

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We reanalysed Yang & Patton's allozyme data, published in Auk in 1981, of Darwin's finches with a variety of distance and cladistic methods to estimate the phylogeny of the group. Different methods yielded different results, nevertheless there was widespread agreement among the distance methods on several groupings. First, the two species of Camarhynrhus grouped near one another, but not always as a monophyletic ,group. Second, Cactospiza pailida and Plagspiza cr~.ctiroslns formed a monophyletic group. Finally, all the methods (including parsimony) supported the monophyly of the ground finches. The three distance methods also found close relationships generally betw~en each of two populations of Geospiza xandms, C. dl/ficilG and G conirostris. There is evidence for inconstancy of evolutionary rates among species. Results from distance methods allowing for rate variation among lineages suggest three conclusions which differ from Yang and Patton's findings. First, the monophyletic ground finches arose from the paraphyletic tree finches. Yang and Patton found that the ground finches and tree finches were sister monophyletic taxa. Second, Ceospzza .scandens appears to be a recently derived species, and not the most basal ground finch. Third, Gfuliginosa is not a recently derived species of ground finch, but was derived from an older split from the remaining ground finches. Most of these conclusions should be considered tentative both because the parsimony trees disagreed sharply with the distance trees and because no clades were strongly supported by the results of bootstrapping and statistical tests of alternative hypotheses. Absence of strong support for clades was probably due to insufficient data.Future phylogenetic studies, preferably using DNA sequence data from several unlinked loci, should sample several populations of each species, and should attempt to assess the importance of hybridization in species phylogeny.