We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assumption on the injectivity radii can be replaced with the request of a controlled growth of the full curvature tensor at order $0$. The control in the assumptions occur via non-necessarily polynomial growth functions. This construction largely extend previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp.
A first main application is to the study of the density property for Sobolev spaces on Riemannian manifolds, namely the problem of guaranteeing the density of smooth compactly supported function in the Sobolev space $W^{k,p}$. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry.
In the particular case $p=2$, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on the space of smooth section of the bundle of $k$-covariant symmetric tensors, we can weaken the assumptions needed to obtain the density property. Namely we prove that the control on the highest order derivative of curvature is not needed in this situation.
Beyond the density property we finally highlight some new applications of our results to disturbed Sobolev inequalities, disturbed $L^{p}$-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian.
The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact support.
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric,…
We study Sobolev-type metrics of fractional order $s\geq0$ on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on $\Diff_c(M)$ vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on $\Diff_c(M)$ is positive for $\dim(M)=1, s>\frac12$ and $\dim(…
We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of the Heisenberg group. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach d…
We improve higher-order CR Sobolev inequalities on $S^{2n+1}$ under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in $C^{2k}(N)$ of higher-order CR Yamabe-type problems when $Y_k(N)<Y_k(\mathbb{H}^n)$.
We establish three families of Sobolev trace inequalities of orders two and four in the unit ball under higher order moments constraint, and are able to construct \emph{smooth} test functions to show all such inequalities are \emph{almost optimal}. Some distinct feature in \emph{almost sharpness} examples between the fourth order and second order Sobolev trace inequalities is discovered. This has been neglected in higher order Sobolev inequality case in \cite{Hang}. As a byproduct, the method of our construction can be used to show the sharpness of the generalized Lebedev-Milin inequality unde…
We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the geodesic distance induced by $W^{s,p}$ vanishes if $sp\le n$ and $s<1$, and is positive otherwise. The proof is achieved by combining the techniques of two recent papers --- [JM19] by the authors, which treated the sub-critical case, and [BHP18] of Bauer, Harms and Preston, which …
We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s > n/p$ or $s \ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conj…
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.
In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and $BV$ spaces in a Carnot group by means of an integral decoupling property of the heat kernel.
In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptions on the target are required. Our proof relies on a careful analysis of the Sobolev spaces of maps involved in the variational process, and on a deformation result which permits to glue a suitable Euclidean end to the geodesic ball.
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on the…
We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to R^n$ of $c$. In this paper we study geometries on the manifold of curves, provided by Sobolev--type metrics $H^j$. We study $H^j$ type metrics for the cases $j=1,2$; we prove estimates, and characterize the completion of the space of smooth curves. As a bonus, we prove that the Fréchet distance of curves (see arXiv:math.DG/0312384) coincides with the dis…
Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geometry with geodesics, gradient flows and curvature. Unfortunately, the simplest such metric induces vanishing path-length distance on shape space. This discovery by Michor and Mumford was the starting point to a quest for stronger, meaningful metrics that should be able to disting…
In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.
We give a new proof of Aubin's improvement of the Sobolev inequality on $\mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $\mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.
We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the Hölder sense. This is done by establishing a general trace inequality on Riemannian manifolds which is an interesting result on its own. We provide examples demonstrating how each of our hypotheses are necessary. In the Appendix by the first author with Christina Sormani, we prove that sequences of compact integral current spaces without boundary (including Riemannian manifolds) that have uniform Hölder bounds on their d…
