Smooth Compact Manifold Research Articles

Topological complexity and efficiency of motion planning algorithms

We introduce a variant of Farber’s topological complexity, defined for smooth compact Riemannian manifolds, which takes into account only motion planners with the lowest possible “average length” of the output paths. We prove that it never differs from topological complexity by more than 1, thus showing that the latter invariant addresses the problem of the existence of motion planners which are “efficient”.

On the isoperimetric quotient over scalar-flat conformal classes

Let (M, g) be a smooth compact Riemannian manifold of dimension n with smooth boundary . Suppose that (M, g) admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) and has a nonumbilic point; or (ii) is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.

Extended Fuller index, sky catastrophes and the Seifert conjecture

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

On probability of high extremes of Gaussian fields with a smooth random trend

Let ξ1≔ξ1(t),t∈Rn be a centered locally ((E,α),Dt)-stationary Gaussian field, ξ2≔ξ2(t),t∈Rn be a centered differentiable in mean-square sense Gaussian field and η≔η(t),t∈Rn particular smooth random field, being independent of ξi, i=1,2. In this paper we discuss the asymptotics of Psupt∈Mk(ξi(t)+η(t))>u,asu→∞, where Mk⊂Rn is a smooth k-dimensional compact manifold, 0<k≤n, i=1,2.

CONFIGURATION CATEGORIES AND HOMOTOPY AUTOMORPHISMS

AbstractLet M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).

Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $$r=1$$ ) and prove that when M is a smooth compact manifold or $$\mathbb {R}^d$$ , the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $$M=\mathbb {R}^d$$ , $$d \ge 2$$ , we also prove that for $$r \ge 2$$ small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order $$\mathscr {O}(c^{2(r-1)})$$ . We also prove a global existence result uniform with respect to c for the NLKG equation on $$\mathbb {R}^3$$ with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on $$\mathbb {R}^3$$ .

Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map.

Equidistribution of the conormal cycle of random nodal sets

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidistributes on the fibres of the cotangent bundle. In the case where the dimension is even, we obtain an upper bound of lower order on the expectation. Using recent results of Alesker, we also deduce some properties on the asymptotic expectation of any smooth valuation including the Euler characteristic of random nodal sets.

Quantitative uniqueness of solutions to parabolic equations

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth Riemannian manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the sharp vanishing order for solutions in term of the C1,1 norm of the potential functions, as well as the L∞ norm of the coefficient functions. Some new quantitative Carleman estimates and three cylinder inequalities are established.

General and refined Montgomery Lemmata

Montgomery’s Lemma on the torus $$\mathbb {T}^d$$ states that a sum of N Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: let (M, g) be a smooth compact d-dimensional manifold without boundary, let $$(\phi _k)_{k=0}^{\infty }$$ denote the Laplacian eigenfunctions, let $$\left\{ x_1, \dots , x_N\right\} \subset M$$ be a set of points and $$\left\{ a_1, \dots , a_N\right\} \subset \mathbb {R}_{\ge 0}$$ be a sequence of nonnegative weights. Then, for all $$X \ge 0$$ , $$\begin{aligned} \sum _{k=0}^{X}{ \left| \sum _{n=1}^{N}{ a_n \phi _k(x_n)} \right| ^2} \gtrsim _{(M,g)} \left( \sum _{i=1}^{N}{a_i^2} \right) \frac{ X}{(\log {X})^{\frac{d}{2}}}. \end{aligned}$$ This result is sharp up to the logarithmic factor. Furthermore, we prove a refined spherical version of Montgomery’s Lemma, and provide applications to estimates of discrepancy and discrete energies of N points on the sphere $$\mathbb {S}^{d}$$ .

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.

Clustered solutions for supercritical elliptic equations on Riemannian manifolds

AbstractLet{(M,g)}be a smooth compact Riemannian manifold of dimension{n\geq 5}. We are concerned with the following elliptic problem:-\Delta_{g}u+a(x)u=u^{\frac{n+2}{n-2}+\varepsilon},\quad u&gt;0\text{ in }M,where{\Delta_{g}=\mathrm{div}_{g}(\nabla)}is the Laplace–Beltrami operator onM,{a(x)}is a{C^{2}}function onMsuch that the operator{-\Delta_{g}+a}is coercive, and{\varepsilon&gt;0}is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has ak-peaks solution for positive integer{k\geq 2}, which blow up and concentrate at one point inM.

Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles

The traceless SU(2) character variety R(S^2,\{a_i,b_i\}_{i=1}^n) of a 2n -punctured 2-sphere is the symplectic reduction of a Hamiltonian n -torus action on the SU(2) character variety of a closed surface of genus n . It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension 4n-6 . For generic holonomy perturbations \pi , the traceless SU(2) character variety R_\pi(Y,L) of an n -stranded tangle L in a homology 3-ball Y is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to R(S^2,\{a_i,b_i\}_{i=1}^n) is a Lagrangian immersion which preserves the cone neighborhood structure near the singular stratum. For generic holonomy perturbations \pi , the variant R_\pi^\natural(Y,L) , obtained by taking the connected sum of L with a Hopf link and considering SO(3) representations with w_2 supported near the extra component, is a smooth compact manifold without boundary of dimension 2n-3 , which Lagrangian immerses into the smooth stratum of R(S^2,\{a_i,b_i\}_{i=1}^n) . The proofs of these assertions consist of stratified transversality arguments to eliminate non-generic strata in the character variety and to insure that the restriction map to the boundary character variety is also generic. The main tool introduced to establish abundance of holonomy perturbations is the use of holonomy perturbations along curves C in a cylinder F\times I , where F is a closed surface. When C is obtained by pushing an embedded curve on F into the cylinder, we prove that the corresponding holonomy perturbation induces one of Goldman's generalized Hamiltonian twist flows on the SU(2) character variety \mathcal{M}(F) associated to the curve C .

$L^p$ almost conformal isometries of Sub-Semi-Riemannian metrics and solvability of a Ricci equation

Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in some sense solvable, without assuming any closeness near a special metric.

Various expansive measures for flows

We discuss a characterization of countably expansive flows in measure-theoretical terms as in the discrete case [2]. More precisely, we define the countably expansive flows and prove that a homeomorphism of a compact metric space is countable expansive just when its suspension flow is. Moreover, we exhibit a measure-expansive flow (in the sense of [4]) which is not countably expansive. Next we define the weak expansive measures for flows and prove that a flow of a compact metric space is countable expansive if and only if it is weak measure-expansive (i.e. every orbit-vanishing measure is weak expansive). Furthermore, unlike the measure-expansive ones, the weak measure-expansive flows may exist on closed surfaces. Finally, it is shown that the integrated flow of a C1 vector field on a compact smooth manifold is C1 stably expansive if and only if it is C1 stably weak measure-expansive.

Asymptotic expansion of the heat kernel trace of Laplacians with polynomial potentials

It is well-known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemmanian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. It is the purpose of this work to show that, in certain circumstances, termwise integration can be used to obtain the asymptotic expansion of the heat kernel trace for Laplace operators endowed with a suitable polynomial potential on unbounded domains. This is achieved by utilizing a resummed form of the asymptotic expansion of the on-diagonal heat kernel.

Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

Points on manifolds with asymptotically optimal covering radius

Given a finite set of points on the Euclidean sphere, the worst case quadrature error for functions in Sobolev spaces has recently been shown to provide upper bounds on the covering radius of the point set. Moreover, sequences of Quasi-Monte Carlo (QMC) designs for Sobolev spaces on the sphere achieve asymptotically optimal covering radii. Here, we extend these results from sequences of QMC designs on the sphere to sequences of weighted QMC designs on compact smooth Riemannian manifolds. We also provide numerical experiments illustrating our findings for the Grassmannian manifold.

Elliptic Complexes on Manifolds with Boundary

We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.

The Friedrichs extension for elliptic wedge operators of second order

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.