Cheeger Type Sobolev Spaces for Metric Space Targets

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above.     

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.     

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let g be a finite-dimensional simple Lie algebra and let Sg be the locally finite part of the algebra of invariants (EndCV⊗Sg(g)) where V is the direct sum of all simple finite-dimensional modules for g and S(g) is the symmetric algebra of g. Given an integral weight ξ, let Ψ=Ψ(ξ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra of Sg and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.     

Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains. S-Y. Cheng’s research partially supported by the CUHK direct grant A/C # 220600260. K. Oden’s research partially supported by the Department of Education Graduate Fellowship     

Normality of products of monotonically normal spaces with compact spaces

Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

On eigenfunctions of Markov processes on trees

We begin by studying the eigenvectors associated to irreducible finite birth and death processes, showing that the i nontrivial eigenvector φ _i admits a succession of i decreasing or increasing stages, each of them crossing zero. Imbedding naturally the finite state space into a continuous segment, one can unequivocally define the zeros of φ _i , which are interlaced with those of φ _i+1. These kind of results are deduced from a general investigation of minimax multi-sets Dirichlet eigenproblems, which leads to a direct construction of the eigenvectors associated to birth and death processes. This approach can be generically extended to eigenvectors of Markov processes living on trees. This enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous Dirichlet eigenproblems and a more general conjecture is presented about related higher order Cheeger inequalities. Finally, we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees.     

Compact differences of composition operators on weighted Dirichlet spaces

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S ², the space of analytic functions whose first derivative is in H ², and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.     

Some local eigenvalue estimates involving curvatures

We first establish a local Faber–Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively.     

Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when ?? > ??₁, the problem has extremal solutions of constant sign and when ?? > ??₂ it has also a nodal (sign-changing) solution. Here ??₁?<???₂ are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p?=?2) we produce two nodal solutions.     

A remark on the stabilization of the 1-d wave equation

We consider the wave equation on an interval of length 1 with an interior damping at ξ and with Dirichlet boundary condition at the two ends. It is well known that, if ξ is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

Cheeger constants, growth and spectrum of locally tessellating planar graphs

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.     

Degree formulae for offset curves

In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S, that is defined depending on a non-empty Zariski open subset of R². The second formula is based on the resultant of the defining polynomial of C, and the polynomial defining generically S. The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C, at their intersection points.     

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class \({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\), we recover isomorphism classes of geometric string structures on Spin _n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class \({\frac{1}{2} p_1}\) together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.     

Vector bundles on a three-dimensional neighborhood of a ruled surface

Let S be a ruled surface inside a smooth threefold W and let E be a vector bundle on a formal neighborhood of S. We find minimal conditions under which the local moduli space of E is finite dimensional and smooth. Moreover, we show that E is a flat limit of a flat family of vector bundles whose general element we describe explicitly.     

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of R^N. The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.     

Some properties of non-compact complete Riemannian manifolds

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold M. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on M, for any q∈(0,∞), every non-negative Lq subharmonic function is constant under a natural decay condition on the Ricci curvature.     

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann problem

In this paper we continue the study started in Hersonsky (in press) [16]. We consider a planar, bounded, m-connected region Ω, and let ∂Ω be its boundary. Let T be a cellular decomposition of Ω∪∂Ω, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f) where S is a special type of a (possibly immersed) genus (m−1)singular flat surface, tiled by rectangles and f is an energy preserving mapping from T(1) onto S. In Hersonsky (in press) [16] the solution of a Dirichlet problem defined on T(0) was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.