Set Of Riemannian Metrics Research Articles

On spectral simplicity of the Hodge Laplacian and curl operator along paths of metrics

We prove that the curl operator on closed oriented 3 3 -manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1 1 -dimensional eigenspaces, even along 1 1 -parameter families of C k \mathcal {C}^k Riemannian metrics, where k ≥ 2 k\geq 2 . We show further that the Hodge Laplacian in dimension 3 3 has two possible sources for nonsimple eigenspaces along generic 1 1 -parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 3 is a meagre codimension 1 1 property with respect to the C k \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 2 property.

Genericity of Nondegenerate Free Boundary CMC Embeddings

Let $$\Sigma ^n$$ and $$M^{n+1}$$ be smooth manifolds with smooth boundary. In this paper, following the techniques developed by White (Indiana Univ Math J 40:161–200, 1991) and Biliotti–Javaloyes–Piccione (Indiana Univ Math J, 1797–1830, 2009), we prove that, given a compact manifold with boundary $$\Sigma ^n$$ and a manifold with boundary $$M^{n+1}$$ , for a generic set of Riemannian metrics on M every free boundary CMC embedding $$\phi :\Sigma \rightarrow M$$ is non-degenerate.

With respect to whom are you critical?

For any compact Riemannian surface S and any point y in S, Qy−1 denotes the set of all points in S for which y is a critical point, and |Qy−1| its cardinality. We proved [2] together with Imre Bárány that |Qy−1|≥1, and that equality for all y∈S characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface S and any point y∈S, the following two main results. There exists an open and dense set of Riemannian metrics g on S for which y is critical with respect to an odd number of points in S, and this is sharp. If S is the torus then |Qy−1|≤5, and if S has genus g≥2 then |Qy−1|≤8g−5. Properties involving points at globally maximal distance on S are eventually presented.

On isospectral compactness in conformal class for 4-manifolds

Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

Uniqueness of solutions of the Yamabe problem on manifolds with boundary

Given a compact manifold with boundary M of dimension m≥3 and a nondegenerate Riemannian metric g∗ having null scalar curvature, constant mean curvature, and unitary volume on the boundary, we show that the set of Riemannian metrics with null scalar curvature, constant mean curvature, and unitary volume on the boundary, near to g∗ is an embedded submanifold of the manifold of all Riemannian metrics on M. Additionally, such submanifold is strongly transversal to the conformal classes. We also prove, using recent results of compactness, that conformal classes of metrics closed to g∗ contain only one of these metrics.

On the set of metrics without local limiting Carleman weights

In the paper [1] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper [7] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of [1], in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is open and dense.

Einstein metrics, harmonic forms, and symplectic four-manifolds

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.

Scalar Curvature and Q-Curvature of Random Metrics

We define a family of probability measures on the set of Riemannian metrics lying in a fixed conformal class, induced by Gaussian probability measures on the (logarithms of) conformal factors. We control the smoothness of the resulting metric by adjusting the decay rate of the variance of the random Fourier coefficients of the conformal factor. On a compact surface, we evaluate the probability of the set of metrics with non-vanishing Gauss curvature, lying in a fixed conformal class. On higher-dimensional manifolds, we estimate the probability of the set of metrics with non-vanishing scalar curvature (or Q-curvature), lying in a fixed conformal class.

A dense G-delta set of Riemannian metrics without the finite blocking property

A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) in M x M, there exists a dense G-delta set of C-infinity Riemannian metrics on M such that (x,y) fails to have the finite blocking property for every g in that set.

Symmetries, non-Euclidean metrics, and patterns in a Swift–Hohenberg model of the visual cortex

The aim of this work is to investigate the effect of the shift-twist symmetry on pattern formation processes in the visual cortex. First, we describe a generic set of Riemannian metrics of the feature space of orientation preference that obeys properties of the shift-twist, translation, and reflection symmetries. Second, these metrics are embedded in a modified Swift-Hohenberg model. As a result we get a pattern formation process that resembles the pattern formation process in the visual cortex. We focus on the final stable patterns that are regular and periodic. In a third step we analyze the influences on pattern formation using weakly nonlinear theory and mode analysis. We compare the results of the present approach with earlier models.

C∞ Genericity of Positive Topological Entropy for Geodesic Flows on S2

We show that there is a C∞ open and dense set of positively curved metrics on S2 whose geodesic flow has positive topological entropy, and thus exhibits chaotic behavior. The geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics. The positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow. Our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior. Very shortly after this manuscript was completed, the authors learned about remarkable recent work by Hofer, Wysocki, and Zehnder [14, 15] on three dimensional Reeb flows. In the special case of geodesic flows on S2, they show that if the geodesic flow has no parabolic closed geodesics (this holds for an open and C∞ dense set of Riemannian metrics on S2), then it possesses either a global surface of section or a heteroclinic orbit. It then immediately follows from the proof of our main theorem that there is a C∞ open and dense set of Riemannian metrics on S2 whose geodesic flow has positive topological entropy. This concludes a program to show that every orientable compact surface has a C∞ open and dense set of Riemannian metrics whose geodesic flow has positive topological entropy.

A class of Riemannian manifolds that pinch when evolved by Ricci flow

The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N n is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and where K is some compact set .