Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

Submersions on open symplectic manifolds

Let M be an open manifold with a symplectic form Ω, and N a manifold with dimN<dimM. We prove that submersions with symplectic fibres satisfy the h-principle. Such submersions define Dirac manifold structures on the given manifold. As an application to this result we show that CPn?CPk−1 admits a submersion into R2(2k−n) with symplectic fibres for n/2<k?n.     

Dirichlet spectrum and heat content

Let M be a complete Riemannian manifold and D⊂M a smoothly bounded domain with compact closure. We use Brownian motion to study the relationship between the Dirichlet spectrum of D and the heat content asymptotics of D. Central to our investigation is a sequence of invariants associated to D defined using exit time moments. We prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions, that our invariants determine the heat content asymptotics associated to the manifold, and that when the manifold is a generic domain in Euclidean space, the invariants determine the Dirichlet spectrum.     

Stability for semilinear elliptic variational inequalities depending on the gradient

In this work we give a result concerning the continuous dependence on the data for weak solutions of a class of semilinear elliptic variational inequalities (P_n) with a nonlinear term depending on the gradient of the solution. This paper can be seen as the second part of the work Matzeu and Servadei (2010) [9], in the sense that here we give a stability result for the C^1,α-weak solutions of problem (P_n) found in Matzeu and Servadei (2010) [9] through variational techniques. To be precise, we show that the solutions of (P_n), found with the arguments of Matzeu and Servadei (2010) [9], converge to a solution of the limiting problem (P), under suitable convergence assumptions on the data.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Foliations by stationary disks of almost complex domains

We study the problem of existence of stationary disks for domains in almost complex manifolds. As a consequence of our results, we prove that any almost complex domain which is a small deformation of a strictly linearly convex domain D⊂Cn with standard complex structure admits a singular foliation by stationary disks passing through any given internal point. Similar results are given for foliations by stationary disks through a given boundary point.     

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Ω?=(−?,?)×S¹, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b?0, near the characteristic set Σ={0}×S¹. We show that the interplay between the order of vanishing of the functions a and b at x=0 plays a role in the Gevrey solvability.     

Topological equivalence of complex polynomials

The following numerical control over the topological equivalence is proved: two complex polynomials in n≠3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions f_s:Cⁿ→C with isolated singularities such that the degree, the number of vanishing cycles and the number of atypical values are constant in the family.     

The wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X^○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y_±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y₊ to Y₋.     

Local structure of ideal shapes of knots

Relatively extremal knots are the relative minima of the ropelength functional in the C¹ topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C1,1 relatively extremal knot in Rn either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. Our main approach is to show that the shortest curves with bounded curvature and C¹ boundary conditions in Rn contain CLC (circle-line-circle) curves, if they do not have constant maximal curvature.     

Infinitely many leaf-wise intersections on cotangent bundles

If the homology of the free loop space of a closed manifold B$B$ is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiberwise star-shaped hypersurfaces in T^∗B$T^{\ast }B$. We illustrate this in the case of the restricted three body problem.     

Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications

In this paper, we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface Σ   in R²ⁿ${\mathbf{R}}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics. As its applications, some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained, and also we prove that if Σ   is C^∞$C^{\infty }$-generic, it carries infinitely many symmetric closed characteristics.     

The Morse–Sard theorem for Clarke critical values

The Morse–Sard theorem states that the set of critical values of a C^k$C^k$ smooth function defined on a Euclidean space R^d${\mathbb{R}}^d$ has Lebesgue measure zero, provided k≥d$k\ge d$. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C^k$C^k$ functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.     

Projective spaces of aC *-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C^*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection =2p–1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.Partially supported by UBACYT TW49 and TX92, PIP 4463 (CONICET) and ANPCYT PICT 97-2259 (Argentina)     

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λ_g: C^∞ (?M) → C^∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λ_gh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If g^t = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λ_g ^t = Λ_g + tΛ_? + o(t). We study the question: How do tensor fields ? look like for which Λ_? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.     

Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).      

Transformations of linear connections,II

We elucidate [9] with two applications. In the first we view connections as differential systems. Specializing this to trivial bundles overS ¹ and applying the theory of Floquet, we obtain equivalent connections with constant Christoffel symbols. In the second application we prove that the canonical connections of parallelizable manifolds (in particular Lie groups) can be obtained from the canonical flat connection of appropriate trivial bundles. Thus, the formalisms of [1], [4], [5] and [6] fit in the general setting of [9].     

Stein流形上(p,q)型Koppelman-Leray-Norguet公式

设M是复n维Stein流形;并设开集D??M具有逐块C¹边界.本文利用陈度量和陈联络,把Stein流形上(0,q)形式的Koppelman-Leray-Norguet公式推广到(p,q)形式,并得到D上?-方程的解.最后,还给出了Stein流形上实非退化强拟凸多面体的Koppelman-Leray-Norguet公式及其?-方程的解.