Curvature Estimates for Critical 4-Manifolds with a Lower Ricci Curvature Bound

We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our previous paper, which worked predominantly on the scale of the curvature radius, the results here provide curvature controls on a fixed scale

Energy and Asymptotics of Ricci-Flat 4-Manifolds with a Killing Field

Given a complete, Ricci-flat 4-manifold with a Killing field, we give an estimate on the manifold's energy in terms of a certain asymptotic quantity of the Killing field. If the Killing field has no zeros and satisfies a certain asymptotic condition, the manifold is flat.Comment: Published Versio

Generalized Kahler Taub-NUTs and Two Exceptional Instantons

We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular we parametrize their geodesics from the origin, determine curvature fall-off rates, volume growth rates for metric balls, and find blow-down limits

Topology of K\"ahler manifolds with weakly pseudoconvex boundary

We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge1$ pseudoconvex boundary components and at least one non-parabolic end, the first betti number of $K$ is at least $l$. In either case, any boundary component has non-vanishing first betti number. If $K$ has one pseudoconvex boundary component with vanishing first betti number, the first betti number of $K$ is also zero. Especially significant are applications to Kahler ALE manifolds, and to Kahler 4-manifolds. This significantly extends prior results in this direction (eg. Kohn-Rossi), and uses substantially simpler methods.Comment: Published Versio

Classification of polytope metrics and complete scalar-flat K\"ahler 4-Manifolds with two symmetries

We study unbounded 2-dimensional metric polytopes such as those arising as K\"ahler quotients of complete K\"ahler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible metrics on on the corresponding K\"ahler 4-manifolds. If the polytope is the plane or half-plane then only flat metrics are possible, and if the polytope has one corner then the 2-parameter family of generalized Taub-NUTs (discovered by Donaldson) are indeed the only possible metrics. Polytopes with $n\ge3$ edges admit an $(n+2)$-dimensional family of possible metrics.Comment: 54 pages, 22 figure

Convergence of compact Ricci solitons

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in dimension 4, where $L^2$ curvature bounds are equivalent to upper bounds on the Euler number. We obtain necessary and sufficient conditions for limits to be compact

Harnack Inequalities for Critical 4-manifolds with a Ricci Curvature Bound

We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating metrics, followed by a geometric/topological triviality result from a previous work.Comment: The third uploaded version corrects some potentially confusing typographical errors. No material adjustments were mad

Moduli spaces of critical Riemannian metrics with L^{n/2} norm curvature bounds

We consider the moduli space of the extremal K\"ahler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on \Riem, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities. Most of our results go through for certain other classes of critical Riemannian metrics.Comment: 72 page

On Conformally Kaehler, Einstein Manifolds

We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.Comment: 45 pages, 9 figures. Simplified proof of Lemma 25. Several minor corrections to section

Spectrogram Feature Losses for Music Source Separation

In this paper we study deep learning-based music source separation, and explore using an alternative loss to the standard spectrogram pixel-level L2 loss for model training. Our main contribution is in demonstrating that adding a high-level feature loss term, extracted from the spectrograms using a VGG net, can improve separation quality vis-a-vis a pure pixel-level loss. We show this improvement in the context of the MMDenseNet, a State-of-the-Art deep learning model for this task, for the extraction of drums and vocal sounds from songs in the musdb18 database, covering a broad range of western music genres. We believe that this finding can be generalized and applied to broader machine learning-based systems in the audio domain.Comment: Accepted for presentation at the 27th European Signal Processing Conference (EUSIPCO 2019