Showing papers in "Communications in Analysis and Geometry in 2012"

Analysis of weighted Laplacian and applications to Ricci solitons

Abstract: We study both function theoretic and spectral properties of the weighted Laplacian ∆f on complete smooth metric measure space (M, g, e dv) with its Bakry-Emery curvature Ricf bounded from below by a constant. In particular, we establish a gradient estimate for positive f−harmonic functions and a sharp upper bound of the bottom spectrum of ∆f in terms of the lower bound of Ricf and the linear growth rate of f. We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of f is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.

On the classification of warped product Einstein metrics

Abstract: The question of when an Einstein metric can be written as a warped product is posed in the text Einstein metrics by Besse. Recently, there have been some interesting results about these spaces found by Kim–Kim, Case–Shu–Wei, and Case. They take the perspective of studying these metrics by studying the equation on the base of the warped product. The resulting equation is similar to the Ricci soliton equation and is a natural equation from the work of Bakry– Emery and their collaborators on curvature dimension inequalities. We will also take this perspective and prove two new results: one is an extension of the work of Kim and Kim and Case-Shu-Wei to manifolds with boundary; the other is a classi…cation result of warped products Einstein metrics over locally conformally ‡at base space.

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Abstract: We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq \Lambda$. Assume that $\Sigma_k$ converges to $\Sigma$ in moduli space, i.e. $\phi_k^\ast(\Sigma_k) \to \Sigma$ as complex structures for diffeomorphisms $\phi_k$. Then we construct a branched conformal immersion $f:\Sigma \to \R^n$ and M\"obius transformations $\sigma_k$, such that for a subsequence $\sigma_k \circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $\Lambda < 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k) \geq \min(8\pi,\omega^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

Cosmetic crossings and Seifert matrices

Abstract: We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.

Scalar curvature and uniruledness on projective manifolds

Abstract: It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold $M$ with a Kahler metric with positive total scalar curvature is uniruled, which is equivalent to every point of $M$ being contained in a rational curve. We also prove that if $M$ possesses a Kahler metric of total scalar curvature equal to zero, then either $M$ is uniruled or its canonical line bundle is torsion. The proof of the latter theorem is partially based on the observation that if $M$ is not uniruled, then the total scalar curvatures of all Kahler metrics on $M$ must have the same sign, which is either zero or negative.

Steady gradient Ricci soliton with curvature in $L^1$

Abstract: A steady gradient Ricci soliton is a triple (Mn, g,∇f) where (Mn, g) is a Riemannian manifold and f is a smooth function on Mn such that Ric = Hess(f). It is said complete if (Mn, g) is complete and the vector field ∇f is complete. By [15], the completeness of (Mn, g) suffices to ensure the completeness of ∇f . In this paper, we prove a rigidity result for steady gradient soliton of nonnegative sectional curvature with curvature in L1(Mn, g).

Schoen–Yau–Gromov–Lawson theory and isoparametric foliations

Abstract: Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation as well. To investigate the topology of the double manifolds, we use K-theory and the representation of the Clifford algebra for the FKM-type, and determine completely the isotropy subgroups of singular orbits for homogeneous case.

Closed twisted products and $\sorth{p}\times \sorth{q}$-invariant special Lagrangian cones

Abstract: We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in C (equivalently special Legendrian or contact stationary submanifolds in S2(p+q)−1) are constructed by combining such objects in C and C using a suitable Legendrian curve in S. We study the geometry of these “twisting” curves and in particular the closing conditions for them. In combination with CarberryMcIntosh’s continuous families of special Legendrian 2-tori [3] and the authors’ higher genus special Legendrians [13], this yields a constellation of new special Lagrangian and Hamiltonian stationary cones in C that are topological products. In particular for all n sufficiently large we exhibit infinitely many topological types of SL and Hamiltonian stationary cone in C which can occur in continuous families of arbitrarily high dimension. A special case of the twisted product construction yields all SO(p) × SO(q)-invariant SL cones in C. These SL cones are higher-dimensional analogues of the SO(2)-invariant SL cones constructed previously by Haskins [8,10] and used in our gluing constructions of higher genus SL cones in C [13]. SO(p)× SO(q)-invariant SL cones play a fundamental role as building blocks in gluing constructions of SL cones in high dimensions [14]. We study some basic geometric features of these cones including their closing and embeddedness properties.

Addendum to “Perelman’s reduced volume and a gap theorem for the Ricci flow”

Abstract: In this addendum to [Yo], we prove that if the Gaussian density of a complete gradient shrinking Ricci soliton is close to that of the Gaussian soliton, then they are isometric to each other. This was shown in [Yo] under the additional assumption that its Ricci curvature is bounded below. We drop this assumption by developing Perelman’s reduced geometry for arbitrary complete gradient shrinking Ricci solitons.

Scalar curvature rigidity with a volume constraint

Abstract: Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.

Existence of Hermitian-Yang-Mills metrics under conifold transitions

Abstract: We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold $\hat{X}$ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in $\hat{X}$. Then under some assumptions we show the existence of Hermitian-Yang-Mills metrics on bundles over a family of threefolds $X_t$ with trivial canonical bundles obtained by performing conifold transitions on $\hat{X}$.

Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces

Abstract: We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.

Stability of coassociative conical singularities

Abstract: We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G 2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on its link. We explicitly calculate the stability index for cones on group orbits.We also describe the stability index for cones fibred by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally, we apply our results to construct the first known examples of coassociative 4-folds with conical singularities in compact manifolds with G 2 holonomy.

On the Neuwirth conjecture for knots

Abstract: Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We also establish the conjecture for all Montesinos knots and for all generalized arborescently alternating knots. For knot exteriors containing closed incompressible surfaces satisfying a simple homological condition, we establish that the knots satisfy the Neuwirth conjecture. If there is a proper degree one map from knot $K$ to knot $K'$ and $K'$ satisfies the Neuwirth conjecture then we prove the same is true for knot $K$. Algorithms are given to decide if a knot satisfies the various versions of the Neuwirth conjecture and also the related conjectures about whether all non-trivial knots have essential surfaces at integer boundary slopes.

Principal bundles over a real algebraic curve

Abstract: Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution \sigma. Let G be a connected complex reductive affine algebraic group, and let \sigma_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to \sigma and \sigma_G. We prove that the points defined over $\mathbb R$ of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these properties and the corresponding property of the underlying holomorphic principal G-bundle is explored. A bijective correspondence between unitary representations and polystable objects is established.

The maximal graph Dirichlet problem in semi-Euclidean spaces

Abstract: The maximal graph Dirichlet problem asks whether there exists a spacelike graph, in a semi-Euclidean space, with a given boundary and with mean curvature everywhere zero. We prove the existence of solutions to this problem under certain assumptions on the given boundary. Most importantly, the results proved here will hold for graphs of codimension greater than 1.

Localization for equivariant cohomology with varying polarization

Abstract: We develop a localization theorem in equivariant cohomology that generalizes both the localization with respect to a component of the momentum map and the localization with respect to the norm square of the momentum map, in that it allows a more flexible polarization. Our proof uses noncompact cobordisms. Applying a case of our formula to symplectic toric manifolds gives the measure-theoretic version of the classical Brianchon-Gram polytope decomposition, thus answering a question posed by Shlomo Sternberg.

Persistently laminar branched surfaces

Abstract: We define sink marks for branched complexes and find conditions for them to determine a branched surface structure. These will be used to construct branched surfaces in knot and tangle complements. We will extend Delman’s theorem and prove that a non 2-bridge Montesinos knot K has a persistently laminar branched surface unless it is equivalent to K(1/2q1, 1/q2, 1/q3, −1) for some positive integers qi. In most cases these branched surfaces are genuine, in which case K admits no atoroidal Seifert fibered surgery. It will also be shown that there are many persistently laminar tangles.

The behaviour of fenchel-nielsen distance under a change of pants decomposition

Abstract: Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex struc- tures on S, which we call the Fenchel-Nielsen Teichmuller space associated to the pair (P,X). This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers (1), (2) and (3), and we compared it to the classical Teichmuller met- ric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel- Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.

Relations in the tautological ring of the universal curve

Abstract: After some background theory we provide a brief summary of what is known about the tautological ring of the moduli space curves. We then formulate a few conjectures about the structure of the tautological ring of the universal curve. These conjectures are analogous to the so-called "Faber conjectures". We verify these conjectures for genus 2 < g < 9. We also study some matrices associated to the conjectures and find a realtionship between these matrices and the corresponding matrices on the moduli space of curves.

Scattering boundary rigidity in the presence of a magnetic field

Abstract: It has been shown in \cite{DPSU} that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in \cite{DPSU} to regions with the same scattering data, where only one is known apriori to be simple. We will then use this results to resolve a local version of a question by Robert Bryant. That is, we show that a surface of constant curvature can not be modified in a small region while keeping all the curves of some fixed constant geodesic curvatures closed.

The Seiberg–Witten equations on manifolds with boundary I: the space of monopoles and their boundary values

Abstract: In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the restriction of the space of monopoles to the boundary is a submersion onto a Lagrangian submanifold of the space of connections and spinors on the boundary. Both these spaces are infinite dimensional, even modulo gauge, since no boundary conditions are specified for the Seiberg-Witten equations on the 3-manifold. We study the analytic properties of these monopole spaces with an eye towards developing a monopole Floer theory for 3-manifolds with boundary, which we pursue in Part II.

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

Abstract: Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space Longzhi Lin Ling Xiao Abstract We define a new modified mean curvature flow (MMCF) in hyperbolic space H n+1 , which interestingly turns out to be the natural negative L 2 -gradient flow of the energy functional intro- duced by De Silva and Spruck in [DS09]. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with prescribed asymptotic boundary at infinity. The proof of our main theorems follows closely Guan and Spruck’s work [GS00], and may be thought of as a parabolic analogue. Keywords. Modified mean curvature flow, Hyperbolic space, Star-shaped hypersurfaces Introduction Let F(z, t) : S n + × [0, ∞) → H n+1 be a one parameter family of complete embedded star-shaped hypersurfaces which are radial graphs in H n+1 over S n + , the upper hemisphere of the unit sphere S n in R n+1 , where the half-space model of H n+1 is used. We say the images Σ t = F(z, t) move by modified mean curvature flow (MMCF) if  ∂ F(z, t) ⊥ = (H − σ) ν , (z, t) ∈ S n × [0, ∞) , H ∂t F(z, 0) = Σ 0 , z ∈ S n + , where H denotes the hyperbolic mean curvature of Σ t , σ ∈ (−1, 1) is a constant, and ν H denotes the outward unit normal of Σ t with respect to the hyperbolic metric. By the half-space model of H n+1 , we mean H n+1 = {(x 0 , x n+1 ) ∈ R n+1 : x n+1 > 0} L. Lin: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; e-mail: lzlin@math.rutgers.edu L. Xiao: Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA; e-mail: lxiao@math.jhu.edu Mathematics Subject Classification (2010): Primary 53C44; Secondary 35K20, 58J35

Parallel and symmetric $2$-tensor fields on pseudo-Riemannian cones

Abstract: We study the complete pseudo-Riemannian manifolds whose cone admits a non-trivial parallel symmetric 2-tensor field. We are able to give an extensive description of these manifolds except when the tensor field is given by a nilpotent endomorphism. In this case, we are able to describe only a dense open subset of them. Moreover, we construct examples with non-constant curvature where this open set is proper.

Derived deformations of schemes

Abstract: We introduce a new approach to constructing derived deformation groupoids, by considering them as parameter spaces for strong homotopy bialgebras. This allows them to be constructed for all classical deformation problems, such as defor- mations of an arbitrary scheme, in any characteristic.

Invariants of the harmonic conformal class of an asymptotically flat manifold

Abstract: Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.

Geometry of singular space

Abstract: This paper grew out from my talk for the inauguration of the Riemann Center in Hanover, Germany. In an attempt to understand what Riemann said in his famous paper in 1854 on the foundation of geometry, I propose a theory of geometry that hopefully can be used to understand singular space that may still satisfy the Einstein equation in a generalized sense. Some calculations are made in the appendix that allow us to perform Hodge theory, to calculate the heat kernel within our abstract framework.

On an efficient induction step with Nklt(X,D) -- notes to Todorov

Abstract: Applying the effective induction on Nklt(X,D) developed by Hacon–McKernan and Takayama, Todorov proved that φ5 is birational for projective 3-folds V with vol(V ) ≫ 0, which was recently improved by Di Biagio in loosing the volume constraint. The observation is that the least efficient induction step can be studied in an alternative way, which allows us to assert Todorov’s statement for vol(V ) > 12. The 4-dimensional analog is also given in this note. The idea works well for all dimensions.