Showing papers in "Communications in Analysis and Geometry in 1993"

Sobolev spaces and harmonic maps for metric space targets

Abstract: When one studies variational problems for maps between Riemannian manifolds one must consider spaces which we denote Vr'(r2,X). Here ft is a compact domain in a Riemannian manifold, X is a second Riemannian manifold, p G [l,oo), and W indicates that the first derivatives of the map are L(0). For p > n such maps will be continuous, and the corresponding space W(Cl^X) can be given the structure of a smooth Banach manifold. This is because, for p > n, any map which is close in W^ distance to a map ^o can be described as a pointwise small deformation of UQ. This linear space of W deformations is then a Banach space on which one can locally model W'(Q^ X). For p < n this is no longer possible, and the definition of the space W>{p,,X) becomes much less clear. This problem was first encountered by C.B. Morrey [Mo] in case n = dimfi = 2 and p = 2. A great deal of effort was spent by Morrey to give a definition of this space. In more recent times people have exploited the embedding theorem of J. Nash, and considered X to be a smooth submanifold of a Euclidean space M^. If we define W'(fi, X) to be the subset of the Banach space VF^f^R^) consisting of those maps with image essentially in X, it turns out that this gives a workable definition for many purposes. An aesthetic drawback of this definition is that the space VF'(J7, X) should depend only on the metric of X and not on the embedding of X into R. A much more serious difficulty arises if one attempts to consider maps to spaces X which are not smooth Riemannian manifolds. These

Evolution of harmonic maps with Dirichlet boundary conditions

Abstract: In this paper we shall study a left over problem concerning the heat flow of harmonic maps on manifolds with boundary. Let (M, g) be a compact smooth m-dimensional Reimannian manifold with nonempty smooth boundary <9M, and let (iV, h) be a compact smooth n-dimensional Reimannian manifold without boundary. We denote M U dM by M. Since (AT, h) can be isometrically embedded into an Euclidean space M, for some k > n, we may view TV as a submanifold of R. In local coordinates on M, the energy of a map u : M —> N ^-> R^ is given by

Monotonicity formulas for parabolic flows on manifolds

Abstract: Recently Michael Struwe [S] and Gerhard Huisken [Hu2] have independently derived monotonicity formulas for the Harmonic Map heat flow on a Euclidean domain and for the Mean Curvature flow of a hypersurface in Euclidean space. In this paper we show how to generalize these results to the case of flows on a general compact manifold, and we also give the analogous monotonicity formula for the Yang-Mills heat flow. The key ingredient is a matrix Harnack estimate for positive solutions to the scalar heat equation given in [H]. In [GrH] the authors show how to use the monotonicity formula to prove that rapidly forming singularities in the Harmonic Map heat flow are asymptotic to homothetically shrinking solitons; similar results may be expected in other cases, as Huisken does in [Hu2] for the Mean Curvature flow in Euclidean space. We only obtain strict monotonicity for a special class of metrics, but in general there is an error term which is small enough to give the same effect. (Chen Yummei and Michael Struwe [CS] give a different approach to the error on manifolds.) The special class of metrics are those which are Ricci parallel (so that DiRjk = 0) and have weakly positive sectional curvature (so that RijkiViWjVkWt > 0 for all vectors V and W). This holds for example if M is flat or a sphere or a complex projective space, or a product of such, or a quotient of a product by a finite free group of isometries. In each case we consider a solution to our parabolic equation on a compact manifold M for some finite time interval 0 < t < T, and we let k be any

Nonuniqueness and high energy solutions for a conformally invariant scalar equation

Abstract: The Yamabe problem asserts that given any compact Riemannian manifold (M,<7), without boundary and of dimension greater than or equal to three, there exists a conformally related metric 3, with constant scalar curvature. A glimpse into the colorful history of this problem is given by [20, 19, 1, 14]. Following Schoen's resolution of the Yamabe problem [14], there has been great effort and success in better understanding the quantity and behavior of metrics which can occur as solutions. In this paper we prove the existence of arbitrarily many distinct solutions to the Yamabe problem with constant positive scalar curvature, all lying within a fixed conformal class which is arbitrarily close, in the C topology, to the conformal class of any given metric of positive scalar curvature. Before stating the result precisely we recall some of what is know about the existence of metrics of constant scalar curvature. For a complete and accessible discussion of the Yamabe problem we refer the reader to the excellent survey article by J. Lee and T. Parker [9]. Constant scalar curvature metrics arise as the critical points of the total scalar curvature functional

Quaternionic reduction and Einstein manifolds

Abstract: A quaternionic Kahler manifold (M,g) is a Riemannian manifold of dimension 4n, where n > 1, whose reduced holonomy group is a subgroup of Sp(n)-Sp(l). One extends this definition to the case when n = 1 by saying a 4 dimensional quaternionic Kahler manifold is one that is both self-dual and Einstein. Ever since this type of geometry was introduced by Ishihara [II] there has been a continuous effort to study and classify these spaces. Wolf [Wo] classified all compact, homogeneous, and symmetric quaternionic Kahler manifolds. This classification was extended to the non-compact case by Alekseevskii [A]. As the group Sp(n)-Sp(l) appears in Berger's [Ber] classification theorem as the possible holonomy group of a locally irreducible, non-symmetric Riemannian space, this naturally raises the question of the existence of nonsymmetric space examples. [A] and [Gl] gave examples of non-compact manifolds with quaternionic Kahler metrics with negative scalar curvature which are not locally symmetric. If (M, g) is compact with positive scalar curvature the classification problem has not been answered in general. There are, however, some classification results in dimension four [H] and eight [PoSal], where the only compact quaternionic Kahler manifolds of positive scalar curvature are, in fact, the symmetric Wolf spaces.

Compactification of minimal submanifolds of hyperbolic space

Abstract: In this paper we study the geometry of complete minimal submanifolds of hyperbolic space HI. Specifically, we are interested in m-dimensional submanifolds whose second fundamental form A satisfies fM \\A\\ m < oo where \\A\\ is the norm of A. To motivate this hypothesis we briefly outline the main results when the ambient space is R. Osserman [15] and Chern-Osserman [3], showed that for a complete minimal immersion (cmi for short) M —> M, with finite total curvature, it is possible to compactify M by the Gauss map g: M —> Gn^ which maps p 6 M to the 2-plane Tp(M). By the Weierstrass representation g is a holomorphic curve in (j?n,2, viewed as the complex quadric Qn-2 = {zl H h z* = 0} of the complex projective plane CP. They showed that when the total curvature C(M) = JM K is finite, M is of finite conformal type, i.e., M is conformally equivalent to a closed surface M with a finite number of points removed, and that g extends holomorphically to M. In particular this implies that the total curvature is quantified by C(M) = 27rA;, k an integer, and that M is properly immersed. For a cmi M—>lR5 m > 2) Anderson [2] has obtained a generalization of the Chern-Osserman result. He proved that |w4|(p) goes to 0, as the distance dfePo) of p to a fixed point po goes to infinity. Using the fact that the class of minimal submanifolds is invariant by the homotheties of R, he proved that \\A\\(p) = IJ<(p)/d(p,po), where /x(p) —> 0 as dfapo) —> oo. Analysing the distance function of R restricted to M he concludes that M is properly immersed and that, outside a compact set, M is transversal to the spheres Sr

Convergence of circle packings of finite valence to Riemann mappings

Abstract: In [R-S], the conjecture by W. Thurston [Th] that the hexagonal circle packings can be used to approximate the Riemann mapping (in the topology of uniform convergence in compact subsets) is proved; and in [He], the derivatives of these approximations are shown to be convergent. We show in Section 1 that the methods used in [R-S] in the case of hexagonal packings can be easily extended to the case of nonhexagonal circle packing with bounded radii ratios. We note that Stephenson had taken the major steps toward such an extension in [Ste]. Although he follows the overall strategy of [R-S], he replaces certain key steps by parabolistic arguments which have an interesting nterpretation in terms of the flow of electricity in a network. In Section 2, we show that the method of [He] can be extended to a more general class of non hexagonal packings. Specifically, the restriction in [Ste] that the radii ratios be bounded can be replaced by the much weaker condition that the circle packings have uniformly bounded valence.

Quasi-convergence of Ricci flow for a class of metrics

Abstract: We show that for a certain family of Riemannian metrics on a twisted three-torus, the Ricci flow always asymptotically approaches that of a sub-family of locally homogeneous metrics.

The Poincaré dual of the Weil–Petersson Kähler two-form

Abstract: We consider a family of conjectures due to Witten which relate the Miller-Morita-Mumford cohomology classes to explicit cycles in a certain cell-decomposition of the moduli space of punctured Riemann surfaces. A version of the first of these conjectures is proved here, and the computational geometric proof leads directly to Rogers' version of the dilogarithm and its Abel-Spence functional equation.

The Dirichlet problem at infinity for nonpositively curved manifolds

Abstract: Let Mn be an n-dimensional complete non-compact Riemannian manifold. The Dirichlet problem of harmonic functions with prescribed boundary data at infinity is a very interesting problem in geometry and analysis. In [S], using a probablistic approach, this problem was solved when M is simply-connected and the sectional curvatures are bounded between two negative constants. Then, following the program of Choi, it was solved in [A] under the same assumptions by a more geometric approach. In this paper, we'll follow the program by Anderson and Schoen in [A-S]. In fact, the more general problem of the Martin boundary was solved in [A-S]. The approach of Anderson and Schoen is as follows: For a given boundary data, a smooth function f on M which assumes the boundary data is constructed explicitly. Then, on a geodesic ball B(x0 , R) with center x 0 and radius R solve

The space of minimal annuli bounded by an extremal pair of planar curves

Abstract: In 1956 Shiffman [14] proved that every minimally immersed annulus in M bounded by convex curves in parallel planes is embedded. He proved this theorem by showing that the minimal annulus was foliated by convex curves in parallel planes. We are able to prove a related embeddedness theorem for extremal convex planar curves. Recall that a subset of R is extremal if it is contained on the boundary of its convex hull. We will call a pair of convex curves extremal if their union is extremal.

Perturbation of domains in the Pompeiu problem

Abstract: An old problem in integral geometry called the Pompeiu problem is closely related to the existence of a solution of the over-determined Neumann problem: (N)λ ( ∆u+ λu = 0 in Ω, ∂u ∂ν = 0, u ≡ constant on ∂Ω. It is easy to see (N)λ holds if Ω is a ball. In this paper we shall give a quantitative estimate of the following statement in terms of one parameter family of domains and some special values of Bessel functions: If Ω is sufficiently ‘close to’ a ball and if (N)λ holds for a bounded λ, then Ω must be a ball. §