Hardy Spaces of Differential Forms on Riemannian Manifolds

Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H ^p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H ^p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ^∞ functional calculus and Hodge decomposition, are given.      

Local Hardy Spaces of Differential Forms on Riemannian Manifolds

We define local Hardy spaces of differential forms $h^{p}_{\mathcal{D}}(\wedge T^{*}M)$ for all p∈[1,∞] that are adapted to a class of first-order differential operators $\mathcal{D}$ on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge–Dirac operator on M and Δ=D ² is the Hodge–Laplacian, then the local geometric Riesz transform D(Δ+aI)^?1/2 has a bounded extension to $h^{p}_{D}$ for all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of $h^{1}_{\mathcal{D}}$ in terms of local molecules is also obtained. These results can be viewed as the localization of those for the Hardy spaces of differential forms $H^{p}_{D}(\wedge T^{*}M)$ introduced by Auscher, McIntosh, and Russ.     

Some Equivalent Norms in Sobolev-Besov Spaces on Symmetric Riemannian Manifolds

In this paper we introduce local means on Riemannian symmetricmanifolds of the noncompact type corresponding to the Laplace-Beltramioperator, and investigate equivalent norms in the Sobolev andBesov spaces defined via these means.     

Isomorphisms of Sobolev Spaces on Riemannian Manifolds and Quasiconformal Mappings

We prove that a measurable mapping of domains in a complete Riemannian manifold induces an isomorphism of Sobolev spaces with the first generalized derivatives whose summability exponent equals the (Hausdorff) dimension of the manifold if and only if the mapping coincides with some quasiconformal mapping almost everywhere.

     

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

带边界Riemann流形上的Sobolev空间等价范数

对具光滑边界αＭ的Ｒｉｅｍａｎｎ流形（Ｍ，ｇ），本文建立了Ｓｏｂｏｌｅｖ空间Ｈ（Ｍ）的等价范数     

On Interpolation in L p Spaces

Mathematical Notes -     

On Saddle Submanifolds of Riemannian Manifolds

Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must intersect. Moreover, if M has positive k -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups _1(M) and ₁(M) is surjective.     

On Compactness in Spaces of Bochner Integrable Functions

We show that a bounded subset K of L ^p (,X) is relatively norm compact if and only if K is p-uniformly integrable, scalarly relatively compact, and either tight or flatly concentrated. The scalar relative compactness can be also replaced by several oscillation criteria.     

Proximal Point Algorithm On Riemannian Manifolds

Abstract

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.     

On the Locally Symmetric and Cosympectic Bochner Flat Manifolds

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Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

On the Calculus of Limiting Subjets on Riemannian Manifolds

In this paper fuzzy calculus rules for subjets of order two on finite dimensional Riemannian manifolds are obtained. Then a second order singular subjet derived from a sequence of efficient subsets of symmetric matrices is introduced. Employing fuzzy calculus rules for subjets of order two and various qualification assumptions based on a second order singular subjet, calculus rules for limiting subjets on a finite dimensional Riemannian manifold are obtianed.     

On Riemannian Foliations over Positively Curved Manifolds

We prove that, under reasonable conditions, odd co-dimension Riemannian foliations cannot occur in positively curved manifolds.     

Subgradient Algorithm on Riemannian Manifolds

The subgradient method is generalized to the context of Riemannian manifolds. The motivation can be seen in non-Euclidean metrics that occur in interior-point methods. In that frame, the natural curves for local steps are the geodesies relative to the specific Riemannian manifold. In this paper, the influence of the sectional curvature of the manifold on the convergence of the method is discussed, as well as the proof of convergence if the sectional curvature is nonnegative.     

Proximal Calculus on Riemannian Manifolds

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M. We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M, as well as differentiability and geometrical properties of the distance function to a closed subset C of M. The first-named author was supported by a Marie Curie Intra-European Fellowship of the European Community, Human Resources and Mobility Programme under contract number MEIF CT2003-500927. The second-named author was supported by BFM2003-06420.     

Trust-Region Methods on Riemannian Manifolds

A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.