Compact Manifold Research Articles

Rigidity for inscribed radius estimate of asymptotically hyperbolic Einstein manifold

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper, we generalize this result to asymptotically hyperbolic (AH) Einstein manifold. We get an upper bound of the relative volume of AH manifold and if we combine it with the recent work of Wang and Zhou, then the rigidity is obtained.

Large Deviations for Small Noise Hypoelliptic Diffusion Bridges on Sub-Riemannian Manifolds

In this paper we study a large deviation principle of Freidlin–Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

On manifolds with nonnegative Ricci curvature and the infimum of volume growth order < 2

Abstract We study open (complete and noncompact) n-manifolds M with nonnegative Ricci curvature, under the condition that any asymptotic cone of M splits off an ℝ k {\mathbb{R}^{k}} factor. In particular, we obtain two rigidity results for open n-manifolds M with Ric M ≥ 0 {\mathrm{Ric}_{M}\geq 0} and the infimum of volume growth order < 2 {<2} . The first result asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product ℝ × N {\mathbb{R}\times N} for some compact manifold N. The second asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n - 1 {n-1} dimensional soul.

Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

In this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.

Divisible convex sets with properly embedded cones

AbstractIn this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

The second Yamabe invariant with boundary

Abstract Let ( M , ∂ M , g ) \left(M,\partial M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y ( M , ∂ M , g ) Y\left(M,\partial M,g) , we define the kth Yamabe invariant with boundary Y k ( M , ∂ M , g ) {Y}_{k}\left(M,\partial M,g) . We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on ( M , ∂ M , g ) \left(M,\partial M,g) by using the Yamabe flow with boundary.

Countable dense homogeneity property of connected compact manifolds

Let X be a connected compact manifold and H(X) denote the set of all homeomorphisms from X onto itself. In this paper we provide another proof for the known result that every connected compact manifolds are n−homogeneous and thereby countable dense homogeneous (CDH). We also show that every compact manifold can be expressed as a disjoint union of countable dense subsets. Additionally, we prove that for every h ∈ H(X), ∃ a partition of X in to countable dense subsets on which it is invariant. Mainly, we discuss about the denseness of HA(X) = { h ∈ H(X) | h : A → A is a homeomorphism } in H(X) whenever A is a subset of X. Finally, we prove that if f is a contraction function on X, then the set of all homeomorphisms that commutes with f is a nowhere dense subgroup of H(X).

Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

A conformal transformation of a semi-Riemannian manifold is essential if there is no conformally equivalent metric for which it is an isometry. For Riemannian manifolds the existence of an essential conformal transformation forces the manifold to be conformally flat. This is false for pseudo-Riemannian manifolds, however compact examples of conformally curved manifolds with essential conformal transformation are scarce. Here we give examples of compact conformal manifolds in signature $(4n+2k,4n+2\ell)$ with essential conformal transformations that are locally conformally pseudo-Kähler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given m∈N∖{0} and a compact Riemannian manifold N, we construct for every map u in the critical Sobolev space Wm/(m+1),m+1(Sm,N), a map U:B1m+1→N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

On positive solutions of the nonlocal Yamabe type equation in compact Riemannian manifold

In this paper we deal with a nonlocal curvature equation involving a critical nonlinearity and a lower order perturbation f on a compact Riemannian N-manifold (M,g) (with or without boundary):(−Δg)psu+a(x)up−1=K(x)ups⁎−1+f(x,u) in Mu>0 in Mu=0 on ∂M where (−Δg)ps is the fractional p-Laplacian on (M,g) for p>1 and the fractional exponent 0<s<1 with ps<N and ps⁎:=NpN−ps. Here a(x) is a non-trivial function in L∞(M) and K is a positive smooth function on M. Using critical point theory on the manifold, we prove the existence of a positive solution under certain assumptions.

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

For a compact Riemannian m-manifold (Mm,g),m&gt;1, endowed with a nontrivial conformal vector field ζ with a conformal factor σ, there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ=ζ¯+∇ρ, where ζ¯ satisfies divζ¯=0, which is called the Hodge vector, and ρ is the Hodge potential of ζ. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on Mm. The first result of this article states that a compact Riemannian m-manifold Mm is an m-sphere Sm(c) if and only if (1) for a nonzero constant c, the function −σ/c is a solution of the Poisson equation Δρ=mσ, and (2) the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2. The second result states that if Mm has constant scalar curvature τ=m(m−1)c&gt;0, then it is an Sm(c) if and only if the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of Sm(c) using the affinity tensor of the Hodge vector ζ¯ of a conformal vector field ζ on a compact Riemannian manifold Mm with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold Mm, m&gt;2, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ.

Nonexistence of energy function for surface diffeomorphisms with horseshoe basic sets

The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent set of the system, which called an energy function, qualitatively determines the dynamics of the system. For continuous-time systems, which models dissipative processes, such a function naturally arises as an energy of the system. The existence of an energy function for arbitrary flows on compact manifolds is known. Examples of Morse–Smale diffeomorphism without energy function exist on n-dimensional manifolds starting with n>2. For systems with chaotic dynamics on surfaces, the existence of energy functions has also been established in cases where the dimension of non-trivial basic sets is greater than or equal to 1. When the chain-recurrent set of a diffeomorphism contains at least one zero-dimensional non-trivial basic set, the question of the existence of the energy function is still open. To date, it is known that the answer is negative if the zero-dimensional non-trivial basic set does not contain pairs of conjugated points. The present paper is devoted to the question of existence of an energy function for systems having a basic set with pairs of conjugated points. The primary outcome of this study is that there is no energy function for a dynamical system containing a Smale horseshoe as a basic set.

On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching

AbstractWe introduce simple conditions ensuring that invariant distributions of a Feller Markov chain on a compact Riemannian manifold are absolutely continuous with a lower semi-continuous, continuous or smooth density with respect to the Riemannian measure. This is applied to Markov chains obtained by random composition of maps and to piecewise deterministic Markov processes obtained by random switching between flows.

Pull-back vector fields' bienergy

This work delves into the bienergy of a pull-back vector field from a Riemannian manifold (M, g) to its tangent bundle TN, which is endowed with the Sasaki metric hs. The central question under investigation is whether such a pull-back vector field exhibits biharmonic properties. Through rigorous proof, we establish that if is a parallel vector field, then the pull-back bundle V ∊ Γ(φ−1(TN)) indeed manifests as biharmonic. This finding is significant, particularly for pull-back vector fields on compact manifolds (M, g) that encompass the harmonic map φ. The implications of this work contribute to a deeper understanding of the interactions between vector fields and harmonic mappings within the geometry of Riemannian manifolds, potentially paving the way for further explorations in the realm of differential geometry and its applications. The results affirm the biharmonic nature of parallel vector fields in the specified context, thereby enriching the theoretical framework surrounding the study of bienergy and harmonic maps.

An index theorem for loop spaces

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact S1-manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss Lt,xqc-estimates on intervals of length log⁡λ⋅λ−1 for initial data whose frequencies are comparable to λ, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity

We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu+(−Δ)α2u=2γβeβ|u|2u on d-dimensional compact Riemannian manifolds Md, for a dispersion parameter α>d, some coupling constant β>0, and γ≠0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ>0, the measure is well-defined in the whole regime α>d and β>0 (Theorem 1.1 (i)), while in the focusing case γ<0 its partition function is always infinite for any α>d and β>0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure, in the defocusing case γ>0, in the whole regime α>d and β>0 (Theorem 1.4 (i)). In the large dispersion regime α>2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β>0 and γ≠0. Using the Gibbs measure, we prove that solutions are almost surely global for any β,γ>0, and that the Gibbs measure is invariant (Theorem 1.4 (ii)). (iii) Finally, in the particular case d=1 and M=T, we are able to exploit some probabilistic multilinear smoothing effects to build a global probabilistic flow for (expNLS) for 1+22<α≤2 for any β,γ>0 (Theorem 1.6).