Showing papers in "Journal of Differential Geometry in 1996"

Compact Riemannian 7-manifolds with holonomy G2. II

Abstract: This is the second of two papers about metrics of holonomy G2 on compact 7manifolds. In our first paper [15] we established the existence of a family of metrics of holonomy G2 on a single, compact, simply-connected 7-manifold M , using three general results, Theorems A, B and C. Our purpose in this paper is to explore the theory of compact Riemannian 7-manifolds with holonomy G2 in greater detail. By relying on Theorems A-C we will be able to avoid the emphasis on analysis that characterized [15], so that this sequel will have a more topological flavour. The paper has four chapters. The first chapter consists of introductory material. Section 1.1 gives some elementary geometric and topological material on compact 7-manifolds with torsion-free G2structures. Then §1.2 describes the holonomy groups SU(2) and SU(3), and §1.3 explains the concept of asymptotically locally Euclidean Riemannian manifolds (shortened to ALE spaces) with special holonomy. Recall that in [15], a compact 7-manifold M was defined by desingularizing a quotient T /Γ of the 7-torus by a finite group of isometries Γ ∼= Z2. The subject of Chapters 2 and 3 is a generalization of this idea. Chapter 2 defines a general construction for compact 7-manifolds with torsion-free G2structures, which works by desingularizing quotients T /Γ for finite groups Γ. The ALE spaces with holonomy SU(2) and SU(3) discussed in §1.3 are an essential ingredient in performing this desingularization. The central result of Chapter 2 is Theorem 2.2.3, which states that given a suitable finite group Γ and certain other data, one may construct a compact 7manifold M from T /Γ that admits torsion-free G2structures. This result is proved using Theorems A-C of [15]. Chapter 3 is devoted entirely to examples of this construction. We give many examples of compact 7-manifolds with holonomy G2, and determine their basic topological invariants — the Betti numbers and fundamental group. Finally, in Chapter 4 we discuss some areas of interest, and give a number of open problems. This paper is not written to be read independently of [15]. The language and results of [15] will be used freely, in particular the introductory material in [15, §1.1]. For reference we reproduce here the model 3and 4forms φ, ∗φ defining the flat G2structure on R, as given in [15, §1.1]:

The symplectic geometry of polygons in Euclidean space

Abstract: We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complex-analytically isomorphic to the weighted quotient of (S 2) n constructed by Deligne and Mostow. We study the Hamiltonian ows on M r obtained by bending the polygon along diagonals and show the group generated by such ows acts transitively on M r. We also relate these ows to the twist ows of Goldman and Jeerey-Weitsman.

Level set approach to mean curvature flow in arbitrary codimension

Abstract: We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d — k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and his connection to the level set solutions is also provided.

Bubble tree convergence for harmonic maps

Abstract: Let Σ be a compact Riemann surface Any sequence fn : Σ — > M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : Σ -> M and a tree of bubbles fk : S2 -> M We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the fn converge pointwise We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps fn when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and {/n} is a Palais-Smale sequence for the harmonic map energy

Global asymptotic limit of solutions of the Cahn-Hilliard equation

Abstract: We study the asymptotic limit, as ε \\ 0, of solutions of the Cahn-Hilliard equation 1 under the assumption that the initial energy is bounded independent of ε. Here / = F', and F is a smooth function taking its global minimum 0 only at u — ± 1 . We show that there is a subsequence of {ιt}o 0, regardless of initial energy distributions.

On a conjecture of Clemens on rational curves on hypersurfaces

Abstract: In [2], H. Clemens proved the following theorem: 0.1 Theorem. Let X C P be a general hypersurface of degree d > 2n — 1. Then X contains no rational curve. In [3],[4] Ein generalized Clemens theorem in two directions; he considered a smooth projective variety M of dimension n, instead of P n (which is a mild generalization since any such M can be projected to P), and general complete intersections I c M o f type (efi,... , dk) and proved: 0.2 Theorem. If dι + ... + dk > 2ra — fc — I + 1, any subvariety Y of X of dimension I has a desingularisation Ϋ which has an effective canonical bundle; if the inequality is strict, the sections of Ky separate generic points of Ϋ. In the case of divisors F c l , this result has been improved by Xu [11],[12], who proved: 0.3 Theorem. Let Y C X be a divisor, Y a desingularization of Y, thenpg{Ϋ) >n-l ifΣdi > n + 2. In [11], he gave more precise estimates for the minimal genus of a curve in a general surface in P. Now these results are not optimal, excepted in the case of divisors. In fact we will prove in the case of hypersurfaces the following improvement of Clemens and Ein's results: 0.4 Theorem. (See 2.10.) Let X C P n be a general hypersurface of degree d > 2n — I — 1, 1 < I < n — 3; then any subvariety Y of X of dimension I has a desingularization Y with an effective canonical bundle; if the inequality is strict, the sections of Ky separate generic points ofY.

Counting pseudo-holomorphic submanifolds in dimension 4

Abstract: The purpose of this article is to describe a certain invariant (called the Gromov invariant) for compact symplectic 4-manifolds which assigns an integer to each dimension 2-cohomology class. Roughly speaking, the invariant counts, with suitable weights, compact, pseudo-holomorphic submanifolds whose fundamental class is Poincare dual to the cohomology class in question. A version of this invariant was introduced originally by Gromov [2] to study the deformation classes of symplectic structures on manifolds with the homology of QP. Subsequently, Ruan [7] extended Gromov's constructions to all symplectic 4-manifolds; the generalization of Ruan counts only connected, pseudo-holomorphic submanifolds. The invariant described below generalizes the construction of Ruan. The definition was sketched in [10] where the invariant was identified with the Seiberg-Witten invariants [13] of the symplectic manifold. However, the definition in [10] is incomplete in one respect in its description of counting weights for multiply covered tori with trivial normal bundle. (The discussion in [7] is erroneous in this regard.) Thus, this article also serves to clear up any confusion stemming from counting these multiply covered tori. Note that the equivalence claimed in [10] between the SeibergWitten invariant and the Gromov invariant as defined here holds for manifolds with b > 1. The details of the proof will appear shortly (see [11], [12]). This article is organized as follows: Section 1 defines the Gromov invariant as a weighted count of pseudo-holomorphic submanifolds. (See

A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis

Abstract: The aim of this paper is to prove the existence of weak solutions to the equation ∆u+u = 0 which are positive in a domain Ω ⊂ R , vanish at the boundary, and have prescribed isolated singularities. The exponent p is required to lie in the interval (N/(N − 2), (N + 2)/(N − 2)). We also prove the existence of solutions to the equation ∆u+ u = 0 which are positive in a domain Ω ⊂ R and which are singular along arbitrary smooth k-dimensional submanifolds in the interior of these domains provided p lie in the interval ((n− k)/(n− k − 2), (n − k + 2)/(n − k − 2)). A particular case is when p = (n + 2)/(n − 2), in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a mixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.

Closed Weingarten hypersurfaces in semi-riemannian manifolds

Abstract: The existence of closed hypersurfaces of prescribed curvature in semi-riemannian manifolds is proved provided there are barriers.

Dehn surgery on arborescent knots

Abstract: A knot K is called an arborescent knot if it can be obtained by summing and gluing several rational tangles together, see [7] or below for more detailed definitions. Recall that a 3-manifold is called a Haken manifold if it is irreducible and contains an incompressible surface. Following Hatcher [14] we say that a 3-manifold M is laminar if it contains an essential lamination. The purpose of this paper is to study Dehn surgeries on arborescent knots, and see which of these surgered manifolds are laminar, Haken, or hyperbolic. There has been some study on these problems for Montesinos knots. Denote by K = K(p1/q1, . . . , pn/qn) a Montesinos knot obtained by gluing rational tangles corresponding to the rational numbers pi/qi together in a cyclic way, see for example [24] for more details. To avoid the trivial case, we always assume that |qi| ≥ 2. We call n the length of K. Oertel [24] showed that if n ≤ 3 then there are no closed essential surfaces in the knot exterior E(K) = S − IntN(K), and if n ≥ 4 and |qi| ≥ 3, then there are incompressible surfaces which remain incompressible after all nontrivial surgeries. Delman [4, 5] studied essential laminations in E(K), the exterior of K, showing that for most Montesinos knots there are essential laminations in E(K) which remain

Determinantal formulas for orthogonal and symplectic degeneracy loci

Abstract: Given a vector bundle V of rank n on a variety X, together with two complete flags of subbundles, there is a degeneracy locus Xw C X for each w in the symmetric group Sn. With suitable genericity hypotheses, the class of Xw in the Chow group of X is given by a double Schubert polynomial in the first Chern classes of the quotient line bundles of the flags [9]. In this note we give similar formulas for corresponding loci when V has an orthogonal or symplectic structure and the flags are isotropic; there is one such locus Xw for each w in the corresponding Weyl group.

Curvature varifolds with boundary

Abstract: We introduce a new class of nonoriented sets in Rk endowed with a generalized notion of second fundamental form and boundary, proving several compactness and structure properties. Our work extends the definition and some results of J. E. Hutchinson [13] and can be applied to variational problems involving surfaces with boundary.

A decomposition of smooth simply-connected $h$-cobordant 4-manifolds

Abstract: We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

Rationality of secondary classes

Abstract: We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is torsion for holomorphic rank two vector bundles $E$ with an integrable connection over a complex projective variety $X$. We prove also the rationality of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds. We give a sharp higher-dimensional Milnor inequality for the volume regulator of all representations to $PSO(1,n)$ of fundamental groups of compact $n$-dimensional hyperbolic manifolds, announced in our earlier paper.