Compact N-dimensional Manifold Research Articles

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature Ric≥n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Ric}\\ge n-1$$\\end{document} has length ≤nπ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le n \\pi .$$\\end{document} This improves the bound 8π(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi (n-1)$$\\end{document} given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Let (M, g) be a compact Riemannian n-dimensional manifold with umbilic boundary It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have \(\partial M\) as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is \(n=6,7,8\), provided that the Weyl tensor is always not vanishing on the boundary.

Compactness for conformal scalar-flat metrics on umbilic boundary manifolds

Let (M,g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well known that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provided n=8 and the Weyl tensor of the boundary is always different from zero, or if n>8 and the Weyl tensor of M is always different from zero on the boundary.

Besse conjecture for compact manifolds with pinched curvature

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

On the full asymptotics of analytic torsion

The purpose of this article is to study the asymptotic expansion of Ray–Singer analytic torsion associated with powers p of a given positive line bundle over a compact n-dimensional complex manifold, as p→∞. Here we prove that the asymptotic expansion contains only the terms of the form pn−ilog⁡p,pn−i for i∈N. For the first two leading terms it was proved by Bismut–Vasserot. We calculate the coefficients of the terms pn−1log⁡p,pn−1 in the Kähler case and thus answer the question posed in the recent work of Klevtsov–Ma–Marinescu–Wiegmann about quantum Hall effect. Our second result concerns the general asymptotic expansion of the analytic torsion for a compact complex orbifold.

On the C k-embedding of Lorentzian manifolds in Ricci-flat spaces

In this paper, we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area and then motivate both the mathematical and physical interests in this problem. We show that any n-dimensional compact Lorentzian manifold (Mn, g), with g in the Sobolev space Hs+3, s>n2, admits an isometric embedding in a (2n + 2)-dimensional Ricci-flat semi-Riemannian manifold. The sharpest result available for these types of embeddings, in the general setting, comes as a corollary of Greene’s remarkable embedding theorems R. Greene [Mem. Am. Math. Soc. 97, 1 (1970)], which guarantee the embedding of a compact n-dimensional semi-Riemannian manifold into an n(n + 5)-dimensional semi-Euclidean space, thereby guaranteeing the embedding into a Ricci-flat space with the same dimension. The theorem presented here improves this corollary in n2 + 3n − 2 codimensions by replacing the Riemann-flat condition with the Ricci-flat one from the beginning. Finally, we will present a corollary of this theorem, which shows that a compact strip in an n-dimensional globally hyperbolic space-time can be embedded in a (2n + 2)-dimensional Ricci-flat semi-Riemannian manifold.

The Steklov spectrum and coarse discretizations of manifolds with boundary

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to M, with constants depending only on the dimension and bounds on curvature and injectivity radius. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of the manifold M and those of its discretization. Some applications to the construction of sequences of surfaces with boundary of fixed length and with arbitrarily large Steklov spectral gap are given. In particular, we obtain such a sequence for surfaces with connected boundary. The applications are based on the construction of graph-like surfaces which are obtained from sequences of graphs with good expansion properties.

On the proof of the thin sandwich conjecture in arbitrary dimensions

In this paper we show the validity, under certain geometric conditions, of Wheeler’s thin sandwich conjecture for higher dimensional theories of gravity. We extend the results for the 3-dimensional case in Phys. Rev. D 48, 3596–3599 (1993) in two ways. On the one hand, we show that the results presented in Phys. Rev. D 48, 3596–3599 (1993) are valid in arbitrary dimensions, and on the other hand, we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n≥3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.

Complex Monge–Ampère equation for measures supported on real submanifolds

Let $$(X,\omega )$$ be a compact n-dimensional Kahler manifold on which the integral of $$\omega ^n$$ is 1. Let K be an immersed real $$\mathcal {C}^3$$ submanifold of X such that the tangent space at any point of K is not contained in any complex hyperplane of the (real) tangent space at that point of X. Let $$\mu $$ be a probability measure compactly supported on K with $$L^p$$ density for some $$p>1$$ . We prove that the complex Monge–Ampere equation $$(dd^c \varphi + \omega )^n=\mu $$ has a Holder continuous solution.

Lower bound of Ricci flow's existence time

Let ( M n , g ) be a compact n-dimensional ( n ⩾ 2 ) manifold with nonnegative Ricci curvature, and if n ⩾ 3 , then we assume that ( M n , g ) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on ( M n , g ) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for n = 3 under Rc ⩾ 0 assumption among others. Combining these results, we proved the short-time existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

Tangential thickness of manifolds

Given two compact n-dimensional manifolds in the smooth, piecewise linear or topological categories, basic results of B. Mazur and others give simple criteria for determining whether their products with Euclidean spaces of sufficiently large dimension are isomorphic in the given category. This paper studies such questions when the dimensions of the Euclidean space do not satisfy such a condition, mainly for topological manifolds homotopy equivalent to lens spaces with odd prime order fundamental groups. In particular, complete information is obtained for homotopy lens spaces in most dimensions when the Euclidean space is even dimensional. The proofs use basic techniques from surgery theory and a variety of results from homotopy theory, including results of Cohen, Moore and Neisendorfer related to finite exponents of certain unstable homotopy groups.

Low-dimensional surgery and the Yamabe invariant

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants \sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for \Lambda>0. We derive explicit lower bounds for \Lambda in dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S n ,Y ) are not dense in the Sobolev space W 1,n (S n ,Y ). On the other hand we show that if a met- ric space Y is Lipschitz (n 1)-connected, then Lipschitz mappings Lip(X,Y ) are dense in N 1,p (X,Y ) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincare inequality.

Convergence of Vector Bundles with Metrics of Sasaki-Type

If a sequence of Riemannian manifolds, X i , converges in the pointed Gromov–Hausdorff sense to a limit space, X ∞, and if E i are vector bundles over X i endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the E i converges in the pointed Gromov–Hausdorff sense to a metric space, E ∞. The projection maps π i converge to a limit submetry π ∞ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to $\mathbb {R}^{k}/G$ , where G, henceforth called the wane group, is a closed subgroup of O(k) that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms μ i =∥⋅∥ i converge to a map μ ∞ compatible with the rescaling in $\mathbb {R}^{k}/G$ and the $\mathbb {R}$ -action on E i converges to an $\mathbb {R}$ -action on E ∞ compatible with the limiting norm. A natural notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the $\mathbb {R}$ -action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Also, necessary conditions for uniqueness of parallel translates are given in terms of the wane groups. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g., when a single compact n-dimensional manifold is rescaled to a point, the limit fiber is $\mathbb {R}^{n}/H$ where H is the closure of the holonomy group of the compact manifold considered. Both these examples have uniqueness of parallel translates. However, examples for non-uniqueness are also produced by looking at isolated conical singularities.

A Varadhan type estimate on manifolds with time-dependent metrics and constant volume

In this paper, we consider a compact n-dimensional manifold M with a time-dependent smooth Riemannian metric g(t) whose volume is constant in t. We give a suitable form of the fundamental solution of the linear parabolic operator Δg(t)−∂∂t, where Δg(t) stands for the time-dependent Laplacian based on g(t). We focus on the short-time behavior of the given fundamental solution, extending Varadhanʼs estimate which holds in the case where the metric is fixed.

Critical sets of eigenfunctions of the Laplacian

We give an upper bound for the (n−1)-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic n-dimensional Riemannian manifolds. This is the analog of a result on the nodal set of eigenfunctions by H. Donnelly and C. Fefferman.

Some Dirichlet problems arising from conformal geometry

We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ricg and scalar curvature R g satisfy This result generalizes Aviles and McOwen's in the scalar curvature case.

TOTAL SCALAR CURVATURE AND EXISTENCE OF STABLE MINIMAL SURFACES

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

Bessel capacities on compact manifolds and their relation to Poisson capacities

A motivation for this paper comes from the role of Choquet capacities in the study of semilinear elliptic partial differential equations. In particular, the recent progress in the classification of all positive solutions of L u = u α in a bounded smooth domain E ⊂ R d was achieved by using, as a tool, capacities on a smooth manifold ∂ E. Either the Poisson capacities (associated with the Poisson kernel in E) or the Bessel capacities (related to the Bessel kernel) have been used. In this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities. (Two capacities are called equivalent if their ratio is bounded away from 0 and ∞.) In the literature Bessel capacities are considered mostly in the space R d . We introduce two versions of Bessel capacities on a compact N-dimensional manifold. A class C ap ℓ , p of equivalent capacities is defined, for ℓ p ⩽ N , on every compact Lipschitz manifold. Another class C B ℓ , p is defined (for all ℓ > 0 , p > 1 ) in terms of a diffusion process on a C 2 -manifold. These classes coincide when both are defined. If the manifold is the boundary of a bounded C 2 -domain E ⊂ R d , then both versions of the Bessel capacities are equivalent to the Poisson capacities.

A simple proof on the non-existence of shrinking breathers for the Ricci flow

Suppose M is a compact n-dimensional manifold, n≥ 2, with a metric g ij (x, t) that evolves by the Ricci flow ∂ t g ij = −2R ij in M× (0, T). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality.