Related papers: A compactness theorem for stable flat $SL(2,\mathb…

A locally metric connection on a smooth manifold $M$ is a torsion-free connection $D$ on $TM$ with compact restricted holonomy group $\mathrm{Hol}_0(D)$. If the holonomy representation of such a connection is irreducible, then $D$ preserves…

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which…

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…

In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped…

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on…

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact…

In this paper we develop the compactness theorem for $\lambda$-surface in $\mathbb R^3$ with uniform $\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for…

In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded…

We prove a rigidity result for certain critical Z/2 eigensections of the Laplacian on S^2 associated to a flat real line bundle determined by a branch-point configuration. More precisely, we show that every minimal non-degenerate critical…

We generalize a Cheeger-M\"uller type theorem for flat, unitary bundles on infinite covering spaces over manifolds-with-boundary, proven by Burghelea, Friedlander and Kappeller arXiv:dg-ga/9510010 [math.DG]. Employing recent anomaly results…

Let $\mathbb{k}$ be a commutative ring with global dimension zero. We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of $\mathbb{k}$. That is, the $\infty$-category of…

This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and…

In this paper, we prove a convergence theorem for sequences of Einstein Yang-Mills systems on $U(1) $-bundles over closed $n$-manifolds with some bounds for volumes, diameters, $L^{2}$-norms of bundle curvatures and $L^{\frac{n}{2}}$-norms…

We construct decomposed spectral covers for bundles on elliptically fibered Calabi-Yau threefolds whose structure groups are S(U(1) x U(4)), S(U(2) x U(3)) and S(U(1) x U(1) x U(3)) in heterotic string compactifications. The decomposition…

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex…

We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the…

In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space.

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…

Exploiting the description of rings of differential operators as Azumaya algebras on cotangent bundles, we show that the moduli stack of flat connections on a curve (allowed to acquire orbifold points) defined over an algebraically closed…

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces…