Positive Codimension Research Articles

Energy decay for a locally undamped wave equation

We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semi-classical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

Algebraic approximation preserving dimension

We prove that each semialgebraic subset of \({\mathbb R}^n\) of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets.

Highly Anisotropic Scaling Limits

We consider a highly anisotropic $d=2$ Ising spin model whose precise definition can be found at the beginning of Section 2. In this model the spins on a same horizontal line (layer) interact via a $d=1$ Kac potential while the vertical interaction is between nearest neighbors, both interactions being ferromagnetic. The temperature is set equal to 1 which is the mean field critical value, so that the mean field limit for the Kac potential alone does not have a spontaneous magnetization. We compute the phase diagram of the full system in the Lebowitz-Penrose limit showing that due to the vertical interaction it has a spontaneous magnetization. The result is not covered by the Lebowitz-Penrose theory because our Kac potential has support on regions of positive codimension.

Sharp polynomial energy decay for locally undamped waves

In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.

Homotopy equivalence for proper holomorphic mappings

We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.

Permutation invariant lattices

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ∈Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio (2002, 2007), which we previously studied in Fukshansky and Sun (2014). Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all τ different from an n-cycle.

Spectrally unstable domains

Let $H$ be a separable Hilbert space, $A_{c}:\mathcal{D}_{c}\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal{D}_{\min}$ be the domain of the closure of $A_{c}$ and $\mathcal{D}_{\max}$ that of the adjoint. Assume that $\mathcal{D}_{\max}$ with the graph norm is compactly contained in $H$ and that $\mathcal{D}_{\min}$ has finite positive codimension in $\mathcal{D}_{\max}$. Then the set of domains of selfadjoint extensions of $A_{c}$ has the structure of a finite-dimensional manifold $\mathfrak{SA}$ and the spectrum of each of its selfadjoint extensions is bounded from below. If $\zeta$ is strictly below the spectrum of $A$ with a given domain $\mathcal{D}_{0}\in\mathfrak{SA}$, then $\zeta$ is not in the spectrum of $A$ with domain $\mathcal{D}\in\mathfrak{SA}$ near $\mathcal{D}_{0}$. But $\mathfrak{SA}$ contains elements $\mathcal{D}_{0}$ with the property that for every neighborhood $U$ of $\mathcal{D}_{0}$ and every $\zeta\in\mathbb{R}$ there is $\mathcal{D}\in U$ such that $\operatorname{spec}(A_{\mathcal{D} })\cap(-\infty,\zeta)\neq\emptyset$. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_{c}$.

Well-rounded equivariant deformation retracts of Teichmüller spaces

In this paper, we construct spines , i.e., \mathrm {Mod}_g -equivariant deformation retracts, of the Teichmüller space \mathcal T_g of compact Riemann surfaces of genus g . Specifically, we define a \mathrm {Mod}_g -stable subspace S of positive codimension and construct an intrinsic \mathrm {Mod}_g -equivariant deformation retraction from mathcal T_g to S . As an essential part of the proof, we construct a canonical \mathrm {Mod}_g -deformation retraction of the Teichmüller space \mathcal T_g to its thick part \mathcal T_g(\varepsilon) when \varepsilon is sufficiently small. These equivariant deformation retracts of \mathcal T_g give cocompact models of the universal space \underline{E}\mathrm {Mod}_g for proper actions of the mapping class group \mathrm {Mod}_g . These deformation retractions of \mathcal T_g are motivated by the well-rounded deformation retraction of the space of lattices in \mathbb R^n . We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.

Line bundle twisted chiral de Rham complex and bound states of D-branes on toric manifolds

In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two-dimensional toric compact manifolds and Calabi–Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in P3. Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in P2 and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on P2 twisted by SL(N) vector bundle with instanton number k is calculated. In all the cases considered we find the infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.

A support theorem for Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of {\em all} Hilbert schemes of points on the curve. Globally, it follows that a family of such curves with smooth relative compactified Jacobian ("moduli space of D-branes'') in an irreducible curve class on a Calabi-Yau threefold will contribute equally to the BPS invariants in the formulation of Pandharipande and Thomas, and in the formulation of Hosono, Saito, and Takahashi.

Generic Transitivity for Couples of Hamiltonians

We study orbits and reachable sets of generic couples of Hamiltonians H 1, H 2 on a symplectic manifold N. We prove that, C k -generically for k large enough, orbits coincide with the whole of N and that the same is true for reachable sets when N is compact. Our results are stated in terms of a strong form of genericity which makes use of the notion of rectifiable subsets of positive codimension in Banach or Frechet spaces.

Some unlikely intersections beyond André–Oort

AbstractAccording to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over $\overline {\mathbb Q}$. Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension ( k − 1 ) n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.

A note on CR mappings of positive codimension

We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k.

Minimizing coincidence in positive codimension

Let f and g be maps between smooth manifolds M and N of dimensions n + m and n, respectively (where m > 0 and n > 2). Suppose that the image (fxg)(M) intersects the diagonal N × N in finitely many points, whose preimages are smooth m-submanifolds inM. The problem of minimizing the coincidence set Coin(f, g) of the maps f and g with respect to these preimages and/or their components is considered. The author’s earlier results are strengthened. Namely, sufficient conditions under which such a coincidence m-submanifold can be removed without additional dimensional constraints are obtained.

On coherent systems of type (n, d, n + 1) on Petri curves

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding “flip” has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill–Noether theory and the stability of kernels of evaluation maps, thus proving Butler’s conjecture in some cases in which it was not previously known.

Nielsen type invariants and the location of coincidence sets in positive codimension

Let two mappings f, g be given between smooth manifolds M, N of different dimensions n + m and n, respectively. We consider the problem of the coincidence set Coin ( f , g ) minimization. Suppose the set Coin ( f , g ) is equal to a finite union of preimages (under f × g ) of diagonal points of the target space N squared, and each preimage is a closed m-submanifold in M. The minimizing coincidence problem may, then, be considered with respect to those preimages. How to coalesce two of them, to move or to remove one of them via homotopies of the mappings f, g? In this paper we give constructive answers to those questions, under additional conditions.

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set $$V \subset {\mathbb{R}}^{n}$$ . A polynomial of this class describes the volume of the set V + tB n as a function of t, where t is a positive number and B n denotes the unit ball in $${\mathbb{R}}^{n}$$ . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold $${\mathcal{M}}$$ , where $${\mathcal{M}}$$ is isometrically embedded with positive codimension in $${\mathbb{R}}^{n}$$ . A Weyl polynomial describes the volume of a tubular neighborhood of its associated $${\mathcal{M}}$$ as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of $${\mathbb{R}}^{n}$$ into $${\mathbb{R}}^{m}$$ for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold. Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Syzygies of modules with positive codimension

A higher syzygy of a module with positive codimension is a maximal Cohen–Macaulay module that plays an important role in Cohen–Macaulay approximation over Gorenstein rings. We show that every maximal Cohen–Macaulay module is a higher syzygy of some positive codimensional module if and only if the ring is an integral domain. Also we discuss the hierarchy of rings with respect to Cohen–Macaulay approximation by codimensions of modules.

Extension of CR maps of positive codimension

We study the holomorphic extendability of smooth CR maps between real analytic strictly pseudoconvex hypersurfaces in complex affine spaces of different dimensions.