Smooth Riemannian Metrics Research Articles

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, {mathscr {S}}. By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric h_{ij} on a three-manifold Sigma foliated by a one-parameter family of two-spheres {mathscr {S}}_rho . It is shown first that we can always choose the shift such that the {mathscr {S}}_rho surfaces form a smooth inverse mean curvature foliation of Sigma . An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the rho -integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the {mathscr {S}}_{rho } level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

The classification of flat Riemannian metrics on the plane

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally occurring, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane—as a subject—is new to the literature. Much of our research focuses on conformal metrics of the form \(e^{2\varphi }g_0\), where \(\varphi : {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a harmonic function and \(g_0\) is the standard Euclidean metric on \({\mathbb {R}}^2\). We find that all such metrics, which we call “harmonic,” arise from Riemann surfaces.

Min\u2013max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,partial M) where 3le (n+1)le 7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C^infty Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If partial M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold

AbstractIt is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form urn:x-wiley:0025584X:media:mana201600523:mana201600523-math-0001for some smooth functions of the total volume . Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.

Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

This article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE 2 ) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate the applicability of our method, by using it to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In SIFHDE 2 -space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. Results are compared with the state-of-the-art in anisotropic surface and volume meshings using several examples and evaluation metrics.

Regularity of intrinsically convex W2,2 surfaces and a derivation of a homogenized bending theory of convex shells

We prove interior regularity for W2,2 isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature.We then derive the Γ-limit of three dimensional nonlinear shells with inhomogeneous energy density, in the bending energy regime. This derivation is incomplete in that it requires an additional technical hypothesis.

Spaces of nonnegatively curved surfaces

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on $S^2$, $RP^2$, and $\mathbb{C}$ equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary manifold.

On locally conformally flat critical metrics for quadratic functionals

The aim of this note is to present some results about critical metrics for quadratic functional \({\mathcal {F}}_{t,s}\) defined on \({\mathcal {M}}_{1}=\{g\in {\mathcal {M}} | \mathrm{Vol}(g)=1\}\), where \({\mathcal {M}}\) is the space of smooth Riemannian metrics on a closed smooth Riemannian manifold \(M^n\). We know that space form metrics are critical point for \({\mathcal {F}}_{t,s}\). We investigate when the converse is true. In particular, we show that locally conformally flat critical metrics with some additional conditions are space form metrics.

Metrics with nonnegative Ricci curvature on convex three-manifolds

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS13], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.

The semiclassical theory of discontinuous systems and ray-splitting billiards

We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. To identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular we prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics.

Local and global well-posedness of the fractional order EPDiff equation on [formula omitted

Of concern is the study of fractional order Sobolev-type metrics on the group of H∞-diffeomorphism of Rd and on its Sobolev completions Dq(Rd). It is shown that the Hs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd) and on the smooth regular Fréchet–Lie group of all H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.

On the number of invisible directions for a smooth Riemannian metric

In this note we give a construction of a C ∞ -smooth Riemannian metric on R n which is standard Euclidean outside a compact set K and such that it has N = n ( n + 1 ) / 2 invisible directions, meaning that all geodesic lines passing through the set K in these directions remain the same straight lines on exit. For example in the plane our construction gives three invisible directions. This is in contrast with billiard type obstacles where a very sophisticated example due to A. Plakhov and V. Roshchina gives 2 invisible directions in the plane and 3 in the space. We use reflection group of the root system A n in order to make the directions of the roots invisible.

Constant scalar curvature metrics on Hirzebruch surfaces

We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our construction is reduced to an ordinary differential equation called Duffing equation. An ODE for Bach-flat metrics on Hirzebruch surfaces with large isometry group is also derived.

The completion of the manifold of Riemannian metrics

We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the $L^2$ metric. The primary motivation for studying this problem comes from Teichmuller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmuller space with respect to a class of metrics that generalize the Weil-Petersson metric.

Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L 2 metric and show that there exists a unique minimal path between any two points. This path is also given explicitly. As an application of these formulas, we show that the metric completion of the manifold of metrics is a CAT(0) space.

Localized gluing of Riemannian metrics in interpolating their scalar curvature

We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extends the Corvino gluing near infinity to non-constant scalar curvature metrics.

Scaling laws for non-Euclidean plates and theW2,2isometric immersions of Riemannian metrics

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ -convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W 2,2 isometric immersion of a given 2 d metric into .

A shape theorem for Riemannian first-passage percolation

Riemannian first-passage percolation is a continuum model, with a distance function arising from a random Riemannian metric in Rd. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability of 1.

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Frechet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.