Showing papers in "Communications in Analysis and Geometry in 2001"

Constant mean curvature surfaces with Delaunay ends

Abstract: In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas’ construction parallels the earlier construction of singular Yamabe metrics by Schoen [18]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the ‘Delaunay surfaces’ are periodic, and the embedded members of this family (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).

Special Lagrangians, stable bundles and mean curvature flow

Abstract: We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Holder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.

Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces

Abstract: In this paper we study curves with constant geodesic curvature in rotationally symmetric complete surfaces. Under monotonicity conditions on the Gauss curvature we classify the closed embedded ones in planes, cylinders, spheres and projective planes. We also distinguish the stable ones, i.e., the second order minima of perimeter while keeping constant the area enclosed. We prove existence and nonexistence of isoperimetric domains, and we show the isoperimetric domains when they exist.

The large scale geometry of nilpotent Lie groups

Abstract: In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Caratheodory metrics and complete metric spaces.

Complex hyperbolic manifolds homotopy equivalent to a Riemann surface

Abstract: We construct isometric actions of fundamental groups of closed Riemann surfaces on the complex hyperbolic plane, which realize all possible values of Toledo’s invariant τ . For integer values of τ these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over closed Riemann surfaces, whose topological type is described in terms of τ . We relate our geometric construction to arithmetic constructions and discuss integrality properties of τ .

Equivariant Seiberg-Witten Floer Homology

Abstract: In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.

Potential theory on Lipschitz domains in Riemannian manifolds: L^P Hardy, and Holder space results

Abstract: This is a continuation of our paper "Boundary layer methods for Lipschitz domains in Riemannian manifolds," [MT]. In that paper we have initiated a program aimed at extending the layer potential theory for the flat-space Laplacian on Lipschitz domains in the Euclidean space to the setting of variable coefficients, and more generally to the context of Lipschitz domains in Riemannian manifolds. We recall the general setting, which will also be in effect in this paper. Let M be a smooth, compact Riemannian manifold, of real dimension dimM = n, with a Riemannian metric tensor, which we assume is Lipschitz. That is, M is covered by local coordinate charts in which the components gjk of the metric tensor are Lipschitz functions. (Actually, in [MT] it was assumed that the metric tensor was of class C; we will extend the results of [MT] to the Lipschitz case in §2 of this paper.) Then the Laplace-Beltrami operator on M is given in local coordinates by

Family Seiberg–Witten invariants and wall crossing formulas

Abstract: In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber structure plays a more prominent role. We derive some wall crossing formulas measuring how the family Seiberg-Witten invariants change from one chamber to another.

Curvature estimates in asymptotically flat manifolds of positive scalar curvature

Abstract: We consider an asymptotically flat Riemannian spin manifold of positive scalarcurvature. An inequality is derived which bounds the Riemann tensor in terms of thetotal mass and quantifies in which sense curvature must become small when the totalmass tends to zero. 1 Introduction Suppose that (M n ,g) is an asymptotically flat Riemannian spin manifold of positive scalarcurvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifoldis always positive, and is zero if and only if the manifold is flat. This result suggests thatthere should be an inequality which bounds the Riemann tensor in terms of the total massand implies that curvature must become small when the total mass tends to zero. In [4]such curvature estimates were derived in the context of General Relativity for 3-manifoldsbeing hypersurfaces in a Lorentzian manifold. In the present paper, we study the problemmore generally on a Riemannian manifold of dimension n≥ 3. Our curvature estimatesthen give a quantitative relation between the local geometry and global properties of themanifold.The main difficulty in higher dimensions is to bound the Weyl tensor (which for n= 3vanishes identically). Our basic strategy for controlling the Weyl tensor can be understoodfrom the following simple consideration. The existence of a parallel spinor in an open setU⊂ Mimplies that the manifold is Ricci flat in U. Thus it is reasonable that by gettingsuitable estimates for the derivatives of a spinor, one can bound all components of theRicci tensor. This method is used in [4], where a solution of the Dirac equation is analyzedusing the Weitzenbo¨ck formula. But the local existence of a parallel spinor does not implythat the Weyl tensor vanishes. This is the underlying reason why in dimension n>3,our estimates cannot be obtained by looking at one spinor, but we must consider a family(ψ

The curvature of minimal surfaces in singular spaces

Abstract: where KΣ is the Gauss curvature of the surface and KM is the sectional curvature of the tangent plane to the surface in the manifold. This shows that the curvature of a minimal surface is less than or equal to that of the ambient space. In this paper, we will show that this fundamental curvature property of minimal surfaces also holds in certain singular spaces. When a smooth surface Σ has a conformal metric with conformal factor λ, it is well known that the Gaussian curvature KΣ is given by the formula KΣ = − 1 2λ 4 log λ.

Finiteness results for Heegaard surfaces in surgered manifolds

Abstract: We demonstrate that for all but a finite number of Dehn fillings on a cusped manifold, the core of the attached solid torus is isotopic into every Heegaard surface for the filled manifold. Furthermore, if the cusped manifold does not contain a closed, non-peripheral, incompressible surface, then after excluding the aforementioned set and those filled manifolds containing incompressible surfaces (also a finite set) every other manifold obtained by Dehn filling contains at most a finite number of Heegaard surfaces that are not Heegaard surfaces for the cusped manifold. It follows that these manifolds contain a finite number of Heegaard surfaces of bounded genera. For each cusped manifold, the excluded manifolds are contained in a finite set that can be determined algorithmically.

Aleksandrov reflection and geometric evolution of hypersurfaces

Abstract: Consider a compact embedded hypersurface Ft in R\" which moves with speed determined at each point by a function F(KI, ... ,Kmt) of its principal curvatures, for 0 < t < T. We assume the problem is degenerate parabolic, that is, that F{ •, t) is nondecreasing in each of the principal curvatures ^i,... ,Kn. We shall show that for t > 0 the hypersurface Tt satisfies local a priori Lipschitz bounds outside of a convex set determined by To and lying inside its convex hull. Our method is the parabolic analogue of Aleksandrov's method of moving planes [Al], [A2], [A3], [A4], [AVo]. The flow of a smooth hypersurface may be generalized to the evolution of a closed set Ft described as the level set of a continuous function ut which satisfies in the viscosity sense a degenerate parabolic PDE defined by F for 0 < t < oo, [ES], [CGG]. It has recently been noted that this levelset flow, even when starting from a smooth hypersurface To, may develop a nonempty interior after the evolving hypersurface collides with itself or develops singularities [BP], [AIC], [AVe], [K]. We shall prove that the same local Lipschitz bounds as in the hypersurface case hold for the inner and outer boundaries of IV As an application, we give some new results about 1/H flow for nonstar-shaped hypersurfaces, which was recently investigated by Huisken and Ilmanen [HI]. We prove existence and asymptotic roundness, in the Lipschitz sense, for \"extended\" viscosity solutions in R. In contrast, the evolving hypersurfaces given in [HI], which were used to prove a version of the Penrose conjecture, are solutions of a non-local variational problem, valid in general asymptotically flat Riemannian manifolds.

Groups quasi-isometric to symmetric spaces

Abstract: We determine thestructureoffinitely generated groupswhich arequasi-isometricto symmetric spaces of noncompact type, allowing Euclidean de Rham factorsIf X is a symmetric space of noncompact type with no Euclidean de Rham fac-tor, and Γ is a finitely generated group quasi-isometric to the product E k ×X,then there is an exact sequence 1 → H → Γ → L → 1 where H contains afinite index copy of Z k and L is a uniform lattice in the isometry group of X. 1 1 Introduction The main result of this paper is the following theorem.Theorem 1.1 Let X be a symmetric space of noncompact type with no Euclidean deRham factor, and let Nil be a simply connected nilpotent Lie group equipped with aleft-invariant Riemannian metric. Suppose that Γ is a finitely generated group quasi-isometric to Nil× X . Then there is an exact sequence 1 −→ H−→ Γ −→ L−→ 1 (1) where H is a finitely generated group quasi-isometric to Nil and L is a uniform latticein the isometry group of X . In particular, when Nilis the trivial group then Γ is a finite extension of a uniformlattice in Isom(X), and when Nil≃ R

Homeomorphisms of 3-manifolds and the realization of Nielsen number

Abstract: The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a homeomorphism realizing the Nielsen number of f , which is a lower bound for the number of fixed points among all maps homotopic to f . The main theorem of this paper proves this conjecture for all orientation preserving homeomorphisms on geometric or Haken 3-manifolds. It will also be shown that on many manifolds all orientation-preserving homeomorphisms are isotopic to fixed point free homeomorphisms. The proof is based on the understanding of homeomorphisms on 2-orbifolds and 3-manifolds. Thurston’s classification of surface homeomorphisms will be generalized to 2-dimensional orbifolds, which is used to study fiber preserving homeomorphisms of Seifert fiber spaces. Homeomorphisms on most Seifert fiber spaces are indeed isotopic to fiber preserving homeomorphisms, with the exception of four manifolds and orientation-reversing homeomorphisms on lens spaces or S. It will also be determined exactly which manifolds have a unique Seifert fibration up to isotopy. This information will be used to deform a homeomorphism to a certain standard map on each piece of the JSJ decomposition, as well as on the neighborhood of the decomposition tori, which will make it possible to shrink each fixed point class to a single point, and remove inessential fixed point classes.

Brill–Noether theory on singular curves and torsion-free sheaves on surfaces

Abstract: Let C be a smooth curve of genus g. Let WJ(C) be the BrillNoether locus of line bundles of degree d and with r+1 independent sections. The expected dimension of WJ(C) is p(r,d) = g — (r + l)(g — d + r). If p(r, d) > 0 then Fulton and Lazarsfeld have proved that WJ(C) is connected. We prove that this is still true if C is a singular irreducible curve lying on a regular surface S with — Ks generated by global sections. We use this result to give a short new proof of the irreducibility of the moduli space of rank 2 semistable torsion-free sheaves (with a generic polarization and low value of C2) on a K3 surface (this result was recently proved by a different method by O'Grady).

On the Albanese map of compact Kahler manifolds with numerically effective Ricci curvature

Abstract: (we refer to [6] for the definition and properties of nef line bundles on compact complex manifolds). This conjecture was recently settled in the projective case by Qi Zhang (see [24]) using some characteristic p methods. We also mention that for projective threefolds, Thomas Peternell and Fernando Serrano improved Zhang's result, and showed that the Albanese map is a submersion (see [21])The aim of the present paper is to give a differential-geometric approach of the problem; our main theorem solvs a particular case which will be specified below. As it can be easily seen by the Aubin-Calabi-Yau theorem, the hypothesis on the Ricci class is equivalent to the existence of a sequence of Kahler metrics (uJk)k>o on X such that

A connected sum construction for complete minimal surfaces of finite total curvature

Abstract: Background. In the study of minimal surfaces, it is of great value to have a large collection of examples for reference and insight. The purpose of this article is to develop a general procedure for constructing complete minimal surfaces of finite total curvature in E. As far as the author knows, most, if not all, of the nontrivial examples of complete minimal surfaces have been found through extensive use of a global version of the Weierstrass representation theorem; one first assumes the existence of a minimal surface with certain properties, makes a good guess at the complex theoretic data, and then determines whether or not the period problem is solved. This method has had great success. (See [N],[DHKW],[HK] for a survey.) But, since this method uses complex data, it is not easy to gain insight into the examples constructed this way. Furthermore, the period problem restricts examples found by this method to those that have a lot of symmetry in general. For instance, it was only in 1980 that the first example of an arbitrary number of catenoidal ends was found [JM]. For these reasons, a more direct and geometric method of constructing complete minimal surfaces of finite total curvature is desirable. In particular, one hopes that a general method will be found to construct complicated ones from simple ones. In this paper, I provide such a procedure by solving a nonlinear P.D.E. with singular perturbation methods. Kapouleas was the first who applied this technique in order to construct complete embedded minimal surfaces of finite total curvature, by desingularizing the circles of intersection of a collection of catenoids and planes with a common axis [K5]. Here, I prove the following.

Harmonic tori and generalised Jacobi varieties

Abstract: This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite type from Euclidean space into Grassmannians and the projective unitary groups. Further, some of these maps will be purely algebraic. For maps into complex projective space the algebraic maps of the plane are always doubly periodic i.e. they yield 2-tori. The classification of all these algebraic maps remains open.

Minimal surfaces in a wedge of a slab

Abstract: In this paper we construct a new family of minimal surfaces in a wedge of a slab, and study its geometrical properties in depth. These surfaces arise as a solution to Plateau's problem for some polygonal non compact boundaries. By using these examples as barriers, we prove that any properly immersed minimal surface in a wedge of angle 0 of a slab, where 6 G [0,7r[, satisfies the convex hull property. Moreover, we obtain some non existence results for properly immersed minimal surfaces with planar boundary.

Some new behaviour in the deformation theory of Kleinian groups

Abstract: We present examples of hyperbolizable 3-manifolds $M$ with the following property. Let $CC(\pi_1(M))$ denote the space of convex co-compact representations of $\pi_1(M)$. We show that for every $K\geq 1$ there exists a representation $\rho$ in $\bar {CC(\pi_1(M))}$ so that every $K$-quasiconformal deformation of $\rho$ lies in the closure of every component of $CC(\pi_1(M))$. The examples $M$ were discovered by Anderson and Canary.

Minimal surfaces with bounded curvature in Euclidean space

Abstract: Let M. be the set of surfaces of bounded curvature that are completely and minimally embedded in Euclidean space. By setting a uniform lower bound for the injectivity radius of the normal bundle of any such surface M € .M, we can show that there is an embedded tubular neighbourhood of constant radius around any M G M. In particular the area growth of M is not more than cubic, and its spherical area growth, as we will prove, not more than linear. This result can be applied to show that M. is compact with respect to convergence on compact sets of the Euclidean space.

Gromov–Witten invariants and rigidity of Hamiltonian loops with compact support on noncompact symplectic manifolds

Abstract: In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian’s method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the techniques are also used to study the topological rigidity of Hamiltonian loops with compact support on a class of noncompact symplectic manifolds.

Energy minimizing maps to piecewise uniformly regular Lipschitz manifolds

Abstract: for every v G iJ(Jl, X), with t;|^n — Mdn i the sense of trace. Whenever a given Dirichlet boundary data g : 30 -> X admits an extension in iJ(0, X), there exists an energy minimizing extension. It is a very interesting question to ask whether such a minimizing map is regular or at least regular off a small closed set. When X is a smooth compact Riemannian manifold without boundary, the problem has been well studied by many people. It was first proven by Schoen-Uhlenbeck ([SU], [SU1]) (see also GiaquintaGiusti [GG]) that minimizing maps are smooth in ft except a closed subset whose Hausdorff codimension is at least 3. Later, their results were generalized to p-energy minimizing maps by Hardt-Lin [HL], Puchs [Fm], and Luckhaus [Ls] for 1 < p < oo. When X is an Alexander space, which has nonpositive curvature, it was first proven by Gromov-Schoen [GS] and then by Korevaar-Schoen [KS], [KS1], Jost [Jl], [J2], and Serbinowski [Stl] that any minimizing map is Lipschitz continuous in O and continuous up to the boundary dQ if the boundary data are Lipschitz continuous. In the thesis [St2], Serbinowski also showed a small energy regularity theorem in case

Lie groups of Fourier integral operators on open manifolds

Abstract: We endow the group of invertible Fourier integral operators on an open}manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodifferential operators and contact transformations on an open manifold of bounded geometry, and gluing those together via a local section.

Infinitely spatially complex solutions of PDE and their homotopy complexity

Abstract: on the whole real axis —oo < x < oo where u = (^1,^2) € R, F(u) is a given smooth non-negative potential which has the gradient F'(u) which is Lipschitzean in u. We consider solutions bounded for all x, t. Such solutions include in particular x-periodic solutions. Our main goal is to describe domains in the function space which are invariant with respect to this equation, and which correspond to stable, spatially chaotic patterns. Existence of such domains may model persistence of spatially chaotic patterns in many natural phenomena. It turns out that the spatial behavior can be very complex, and the complexity can be described in terms of appropriate algebraic-topological notions. In the scalar case when u = -ui, constant solutions u — m^ where m = const satisfies the equation F{rn) = 0, play an important role. Solutions mi of this equation divide the real line R into a number of intervals 0; = {u : ra; < u < rai+i}. If initial data tx(rr, 0) of (1.1) take value in only one of these intervals, then it is well known that the Maximum Principle implies that the values u(x,t) stay in this interval for alH > 0 (see [13]). Hence, a geometrical partition of R explicitly defines a number of invariant domains in the function space of initial data of (1.1). In the case of the two component system we consider here, the Maximum Principle is not applicable; the topology of the plane differs from the topology of the straight line. Nevertheless, it is possible to define geometrically domains in the function space that are invariant under (1.1). Now an important role is played not by constants, but by special time-independent periodic solutions of (1.1) which correspond to minimal cycles of a corresponding Jacobian metric; the cycles determine \"holes\" in the plane. These holes create a \"soft obstacle\" for dynamics (see Remark

On Brittenham's theorem

Abstract: Mark Brittenham proved [Br 2] the above result under the additional hypothesis that A is transversely oriented. In this note we show how to remove the transverse orientability hypothesis. Removing that hypothesis is of considerable interest, since most known constructions of essential laminations on non Haken manifolds yield non transversely orientable laminations. Brittenham's theorem may play a crucial role in the resolution of the following well known Topological Rigidity Conjecture for laminar manifolds. See [Br2] or [G3] for more details.