1.
The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity - Meeks III, William H.; Pérez, Joaquín; Ros, Antonio
Let \mathcal M be the space of properly embedded minimal surfaces in ?3 with genus zero and two limit ends, normalized so that every surface M ? \mathcal M has horizontal limit tangent plane at infinity and the vertical component of its flux equals one. We prove that if a sequence {M(i)}i ? \mathcal M has the horizontal part of the flux bounded from above, then the Gaussian curvature of the sequence is uniformly bounded. This curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences, concerning the geometry of any properly embedded...
2.
On the Sobolev Space of Isometric Immersions - Pakzad, Mohammad Reza
We prove that every W2,2 isometric immersion from a convex regular domain of ?2 into ?3 can be approximated in W2,2-norm by smooth isometric immersions from the same domain into ?3.
3.
Integrability of Poisson Brackets - Crainic, Marius; Fernandes, Rui Loja
We discuss the integration of Poisson brackets, motivated by our recent solution to the integrability problem for general Lie brackets. We give the precise obstructions to integrating Poisson manifolds, describing the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas, improving on results of Alcalde Cuesta and Hector. We apply our results (and our point of view) to decide about the existence of complete symplectic realizations, to the integrability of submanifolds of Poisson manifolds, and to the study...
4.
Signature Quantization - Guillemin, Victor; Sternberg, Shlomo; Weitsman, Jonathan
We associate to the action of a compact Lie group G on a line bundle over a compact oriented even-dimensional manifold a virtual representation of G using a twisted version of the signature operator. We obtain analogues of various theorems in the more standard theory of geometric quantization. Some of these results were announced in Guillemin, Sternberg and Weitsman, 2003.
6.
Homogeneous Codimension One Foliations on Noncompact Symmetric Spaces - Berndt, Jürgen; Tamaru, Hiroshi
We determine the isometric congruence classes of homogeneous Riemannian foliations of codimension one on connected irreducible Riemannian symmetric spaces of noncompact type. As an application we show that on each connected irreducible Riemannian symmetric space of noncompact type and rank greater than two there exist noncongruent homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.
7.
Exotic Negatively Curved Structures on Cayley Hyperbolic Manifolds - Aravinda, C.S.; Farrell, F.T.
We construct examples of closed negatively curved manifolds M which are homeomorphic but not diffeomorphic to Cayley locally symmetric spaces. Given ? > 0, we can construct such an M with sectional curvatures all in [?4 ? ?,?1].
8.
Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations - Alesker, Semyon
We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.
9.
Combinatorial Ricci Flows on Surfaces - Chow, Bennett; Luo, Feng
We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.
10.
A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature - Gursky, Matthew J.; Viaclovsky, Jeff A.
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples...
11.
Holomorphic De Rham Cohomology of Strongly Pseudoconvex CR Manifolds with S1-actions - Luk, Hing Sun; Yau, Stephen S.-T.
In this paper, we study the holomorphic de Rham cohomology of a compact strongly pseudoconvex CR manifold X in ?N with a transversal holomorphic S1-action. The holomorphic de Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when X is of real dimension three and the Kohn-Rossi cohomology is infinite dimensional. In Theorem A, we relate the holomorphic de Rham cohomology Hkh(X) to the punctured local holomorphic de Rham cohomology at the singularity in the variety V which X bounds. In case X is of real codimension three in ?n+1, we prove that Hn?1h(X) and Hnh(X) have...
12.
L'invariant ? Pour Les Variétés Lipschitziennes - Hilsum, Michel
The ?-invariant has been defined for C?-manifolds by M.F. Atiyah, V.K. Patodi and I.M. Singer, and more recently for manifolds with corners by A. Hassell, R. Mazzeo and R.B. Melrose, and for stratified PL manifolds by H. Moscovici and F.B. Wu. In the present work, this invariant is generalized in the framework of lipschitz riemannian manifolds. This involves selfadjoint extensions of the signature operator on a lipschitz manifold with boundary, and measurable differential forms which represent the Pontryagyn classes of the manifold. This allows us to extend from smooth to topological manifolds the Atiyah-Patodi-Singer index theorem for flat bundles.
13.
Distribution of Resonances for Asymptotically Euclidean Manifolds - Wunsch, Jared; Zworski, Maciej
In this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds, a class of manifolds generalizing Euclidian n-space. Subject to the basic assumption of analyticity near infinity, we show that resolvent of the Laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum. This involves a geometric interpretation of the complex scaling method in terms of deformations in the Grauert tube of the manifold. We then show that the number of resonances (poles of the meromorphic continuation of the resolvent) in a conic neighborhood of ?+ of absolute...
14.
On the Classification of Tight Contact Structures II - Honda, Ko
We present complete classification results for tight contact structures on two classes of 3-manifolds: (i) torus bundles which fiber over the circle and (ii) circle bundles which fiber over closed surfaces.
15.
Analysis on Groups of Diffeomorphisms of Manifolds with Boundary and the Averaged Motion of a Fluid - Shkoller, Steve
We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ? 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the Hs Hilbert-class topologies for s > (n/2) + 1, these subgroups are C? differential manifolds. We consider these new diffeomorphism groups with an H1-equivalent right invariant metric, and prove the existence of unique smooth geodesics ?(t, ·) of this metric, as well as existence and uniqueness of the Jacobi equations associated to...
16.
Moduli of Sheaves on Surfaces and Action of the Oscillator Algebra - Baranovsky, Vladimir
This paper gives a generalization of some results on Hilbert schemes of points on surfaces. Let MG(r,n) (resp. MU(r,n)) be the Gieseker (resp. Uhlenbeck) compactification of the moduli spaces of stable bundles on a smooth projective surface. We show that, for surfaces satisfying some technical condition:
¶ (a) The natural map MG(r,n) ? MU(r,n) generalizing the Hilbert-Chow morphism from the Hilbert scheme of n points on S to the n-th symmetric power, is strictly semi-small in the sense of Goresky-MacPherson with respect to some stratification.
¶ (b) Let Pt(X) be the Intersection Homology Poincare polynomial of X. Generalizing the computation due to...
17.
Compact Self-Dual Manifolds with Torus Actions - Fujiki, Akira
We show that a compact self-dual four-manifold with a smooth action of a two-torus and with non-zero Euler characterestic is necessarily diffeomorphic to a connected sum of copies of complex projective planes, and furthermore the self-dual structure is isomorphic to one of those constructed by Joyce in [11]. This settles a conjecture of Joyce [11] affirmatively. Our method of proof is to show, by complex geometric techniques, that the associated twistor space, which is a compact complex threefold with the induced holomorphic action of algebraic two-torus, has a very special structure and is indeed determined by a certain invariant which...
18.
On Transversally Simple Knots - Birman, Joan S.; Wrinkle, Nancy C.
This paper studies knots that are transversal to the standard contact structure in ?3, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type \mathcal T K is transversally simple if it is determined by its topological knot type \mathcal K and its Bennequin number. The main theorem asserts that any \mathcal T K whose associated \mathcal K satisfies a condition that we call exchange reducibility is transversally simple.
¶ As a first application, we prove that the unlink is transversally simple, extending the main theorem in [10]. As a second application...
19.
Quaternionic Maps Between Hyperkähler Manifolds - Chen, Jingyi; Li, Jiayu
Quaternionic maps (Q-maps) between hyperkähler manifolds are quaternionic analogue of Cauchy-Riemann equations between Kähler manifolds. We provide a necessary and sufficient condition on when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkähler 2-spheres, and give examples of Q-maps which cannot be holomorphic. When the domain is real 4-dimensional, we analyze the structure of the blow-up set of a sequence of Q-maps, and show that the singular set of a stationary Q-map is \mathcal H1-rectifiable.
20.
Higher Type Adjunction Inequalities in Seiberg-Witten Theory - Ozsváth, Peter; Szabó, Zoltán
In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds. These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for four-manifolds which bound circle-bundles over Riemann surfaces.
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