Abstract. The moduli space M of self-dual SU(2)-connections of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite Riemannian 4-manifold (M,g) carries a natural L2 metric g. We prove that, under a certain condition on the scalar curvature and anti-self-dual part of the Weyl tensor of (M,g), the sectional curvature of (M,g) is bounded above and below near the boundary of the completion M and that certain components of the curvature--those tangent to submanifolds consisting of connections of a fixed scale size--are continuous up to the boundary. We observe a curious cancellation phenomenon without which the curvature would blow up at the boundary. We also develop a new method for constructing parametrices for certain Green operators, and refine an estimate of Donaldson concerning the decay rate of curvatures of concentrated instantons.
Abstract. The theorems of this paper give a variety of sharp pointwise upper bounds on the curvatures of (anti-)self-dual connections (instantons) and general Yang-Mills connections. In the instanton case we obtain bounds that depend only on the centers and scales of concentrated connections. These bounds yield new results about the geometry of moduli space near its boundary.
Abstract.
We discuss the geometry of the Yang-Mills configuration spaces and moduli spaces with respect to the L2 metric. We also consider an application to a de Rham-theoretic version of Donaldson's μ-map.
Abstract. This paper is the sequel to Curvature of Yang-Mills moduli spaces near the boundary, I.Moduli spaces M of self-dual SU(2) connections (``instantons'') over a compact Riemannian 4-manifold (M,g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson's compactification; in fact the boundary of the completion is an isometric copy of (M,4(pi)2g) (Groisser and Parker, 1989). In this paper we show that the boundary is, furthermore, a totally geodesic submanifold of the completion. Along the way, we prove a regularity theorem: the continuous extension of g to the ``collar region'' of M is C 1,alpha (in the conventional scale/center coordinates) for small, positive alpha. The proofs rely on some new weighted Sobolev inequalities for concentrated instantons, in which the only dependence of the Sobolev constants on the connection is through the concentration parameter lambda. The exponent in the weighting function translates into the Holder exponent in the regularity theorem.
Abstract. The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate.By associating to an instanton its energy density, we can examine the information metric g on the moduli spaces M of self-dual connections over Riemannian 4-manifolds. Compared with the more widely known L2 metric, the information metric better reflects the conformal invariance of the self-dual Yang-Mills equations, and seems to have better completeness properties. In the case of SU(2) instantons on S4 of charge one, g is known to be the hyperbolic metric on the five-ball. We show more generally that for charge-one SU(2) instantons over 1-connected, positive-definite manifolds, g is nondegenerate and complete in the collar region of M, and is `asymptotically hyperbolic' there; g vanishes at the cone points of M. We give explicit formulae for the metric on the space of instantons of charge one on CP2.
Abstract. We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. Using the geometric representative of &mu(pt.) defined in [S], the boundary region of moduli space contributes 6/64 of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the (k+1)st ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the k-th moduli space. This is peculiar, since the only known embeddings of the k-th moduli space into the k+1st involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region.When using the natural de Rham representatives of &mu(pt.) considered by Witten [W], the boundary region contributes 1/8 of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is surprising but not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.
[S] Sadun, L., A simple geometric representative for μ of a point, Commun. Math. Phys. 178 (1996), 107-113.
[W] Witten, E., Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353.
Abstract. A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison is developed in this paper. The proposed correspondences, called bimorphisms, are more general than those obtained from one-to-one functions. Their topology is investigated in detail.A new criterion for non-rigid shape comparison using bimorphisms is also proposed. The criterion avoids many of the mathematical problems of previous approaches by comparing shapes non-rigidly from the bimorphism.
Geometric invariants are calculated for curves whose shapes can be exactly matched with a bimorphism. The invariants are related to the concave and convex segments of a curve and provide justification for parsing the curve into such segments.
Abstract. We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a "nonlinear averaging algorithm" that computes the average of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold; in particular this applies to averaging a finite list of points. We use this Riemannian averaging algorithm to provide a constructive proof of Karcher's theorem on the existence and local uniqueness of the center of mass, under a somewhat stronger requirement on the support of the distribution. An immediate corollary is a proof of convergence, for a fairly large open set of initial conditions, of the "GPA algorithm" used in many statistical applications to average points in a shape-space. We estimate the convergence rate of our general algorithm and the more special Riemannian averaging algorithm; the results can be used to explain quantitatively the rapid convergence of the GPA algorithm seen in practice.We also show that a mass distribution of small enough support in a Riemannian manifold has a unique center of mass contained in the (suitably defined) convex hull of this support, and use this to define the "primary" center of mass of such a distribution.
Abstract. Euclidean ``(size-and-)shape spaces'' are spaces of configurations of points in RN modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on Generalized Procrustes Analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [G] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.[G] Groisser, D., Newton's Method, Zeroes of Vector Fields, and the Riemannian Center of Mass, Adv. in Appl. Math 33 (2004), 95-135.
Abstract. In [CTDF] an approach is given for minimizing certain functionals on certain spaces N=Maps(Ω, N), where Ω is a domain in some Euclidean space and N is a space of square matrices satisfying some extra condition(s), e.g. symmetry and positive-definiteness. The approach has the advantage that in the associated algorithm, the preservation of constraints is built in automatically. One practical use of such an algorithm its its application to diffusion-tensor imaging, which in recent years has been shown to be a very fruitful approach to certain problems in medical imaging. The method in [CTDF] is motivated by differential-geometric considerations, some of which are discussed briefly in [CTDF] and in greater detail in [CTDF2]. We describe here certain geometric aspects of this approach that are not readily apparent in [CTDF] or [CTDF2]. We also discuss what one can and cannot hope to achieve by this approach.[CTDF] C. Chefd'hotel, D. Tschumperlé, R. Deriche, and O. Faugeras, Constrained flows of matrix-valued functions: application to diffusion tensor regularization, Proceedings of International ECCV 2002 Workshop (Copenhagen), Springer-Verlag, Berlin (2002), 251-265.
[CTDF2] C. Chefd'hotel, D. Tschumperlé, R. Deriche, and O. Faugeras, Regularizing flows for constrained matrix-valued images, J. Math. Imaging and Vision 20 (2004, 147-162.
Abstract. In [TOG] a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least C2. A class of objective functionals, depending on a cost-function Γ, was introduced on the space of bimorphisms between two fixed curves C1 and C2, and it was proposed that one define a "best non-rigid match" between C1 and C2 by minimizing such a functional. In this paper we prove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for 1< j < ∞, the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for C∞ curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fré manifold, not a Banach manifold. This paper lays the groundwork for [G], where we use the Nash Inverse Function Theorem and our results on C∞ curves and bimorphisms to show that if Γ is strongly convex, if C1 and C2 are C∞ curves whose shapes are not too dissimilar (Cj-close for a certain finite j) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.[TOG] Tagare, H.D., O'Shea, D., and Groisser, D., Non-Rigid Shape Comparison of Plane Curves in Images, J. Math. Imaging and Vision 16 (2002), 57-68.
[G] Groisser, D., Certain optimal correspondences between plane curves II: existence, local uniqueness, regularity, and other properties Trans. Amer. Math. Soc., to appear.
- Groisser, D., Certain Optimal Correspondences Between Plane Curves II: Existence, Local Uniqueness, Regularity, and Other Properties, Trans. Amer. Math. Soc., to appear.
Abstract. This paper is a companion to [G], in which several theorems were proven concerning the nature, as infinite-dimensional manifolds, of the shape-space of plane curves and of spaces of certain curve-correspondences called bimorphisms. In [TOG] a class of objective functionals, depending on a choice of cost-function Γ, was introduced on the space of bimorphisms between two fixed curves C1 and C2, and it was proposed that one define a "best non-rigid match" between C1 and C2 by minimizing such a functional. In this paper we use the Nash Inverse Function Theorem to show that for strongly convex functions Γ, if C1 and C2 are C∞ curves whose shapes are not too dissimilar (specifically, are Cj-close for a certain finite j), and neither is a perfect circle, then the minimum of a certain regularized objective functional exists and is locally unique. We also study certain properties of the Euler-Lagrange equation for the objective functional, and obtain regularity results for "exact matches" (bimorphisms for which the objective functional achieves its absolute minimum value of 0) that satisfy a genericity condition.[G] Certain optimal correspondences between plane curves I: Manifolds of shapes and bimorphisms, Trans. Amer. Math. Soc., to appear.
[TOG] Tagare, H.D., O'Shea, D., and Groisser, D., Non-Rigid Shape Comparison of Plane Curves in Images, J. Math. Imaging and Vision 16 (2002), 57-68.