N-dimensional Riemannian Manifold Research Articles

Influence measures on corrected score estimators in functional heteroscedastic measurement error models

This paper deals with the local influence assessment of the effects of minor perturbations of data on corrected score estimators in the functional heteroscedastic measurement error models with known variances. By extending to the context of measurement error models the differential-geometrical framework proposed by Zhu et al. [H.T. Zhu, J.G. Ibrahim, S. Lee, H. Zhang, Perturbation selection and influence measures in local influence analysis, The Annals of Statistics 35 (2007) 2565–2588], an n-dimensional Riemannian manifold is defined. The associated metric tensor is utilized for the selection of appropriate perturbation schemes. The Levi-Civita connection and first and second derivatives of the corrected score estimators are used to construct influence measures. Simple formulas are obtained under different perturbation schemes. A comparison with the slope and curvature based diagnostics defined from the surface of the corrected score estimators formed by perturbation is included. A real data application and a simulated example illustrate the performance of the proposed diagnostics.

An elliptic variational problem involving level surface area on Riemannian manifolds

We study a variational problem involved with a Dirichlet integral and the area of a level surface on arbitrary n-dimensional Riemannian manifolds. We prove optimal regularity results for minimizers and derive a jumping condition along the level surface. We also obtain smoothness of the interface up to a small singular set of Hausdorff dimension less then or equal to n− 8.

Some notes concerning Riemannian metrics of Cheeger Gromoll type

Let (M,g) be an n-dimensional Riemannian manifold and TM its tangent bundle. The purpose of the present paper is three-fold. Firstly, to study paraholomorphy property of two Riemannian metrics ga and ga,b of Cheeger Gromoll type depending on one parameter and two parameters by using compatible paracomplex structures Ja and Ja,b on the tangent bundle TM. Secondly, to classify Killing vector fields on the tangent bundle TM equipped with the Riemannian metric ga,b. Finally, to give a detailed description of geodesics on the tangent bundle TM with respect to the Riemannian metric ga,b.

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.

Conformal Vector Fields and Eigenvectors of Laplacian Operator

In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere Sn(λ).

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let \({(\mathcal{M}, \tilde{g})}\) be an N-dimensional smooth compact Riemannian manifold. We consider the problem \({\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}\), where \({\varepsilon > 0}\) is a small parameter and V is a positive, smooth function in \({\mathcal{M}}\). Let \({\kappa \subset \mathcal{M}}\) be an (N − 1)-dimensional smooth submanifold that divides \({\mathcal{M}}\) into two disjoint components \({\mathcal{M}_{\pm}}\). We assume κ is stationary and non-degenerate relative to the weighted area functional \({\int_{\kappa}V^{\frac{1}{2}}}\). For each integer m ≥ 2, we prove the existence of a sequence \({\varepsilon = \varepsilon_\ell \rightarrow 0}\), and two opposite directional solutions with m-transition layers near κ, whose mutual distance is \({{\rm O}(\varepsilon | \log \varepsilon | )}\). Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.

Remarks on the curvature behavior at the first singular time of the Ricci flow

@@ t gi jD 2Ri j; t2T0; T/; on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T , we show that suitable blow-ups of the evolving metrics converge in the pointed Cheeger‐Gromov sense to a Gaussian shrinker by using Perelman’s W-functional. If the flow has uniformly bounded scalar curvature and develops Type II singularities at T , we show that suitable scalings of the potential functions in Perelman’s entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a Type II singularity at T or it can be smoothly extended past time T .

Multiple existence of solutions for a singularly perturbed nonlinear elliptic problem on a Riemannian manifold

Let (M, g) be a compact smooth N-dimensional Riemannian manifold without boundary. We consider the multiple existence of positive solutions of the problemwhere Δg stands for the Laplacian in M and f ε C2(M).

Complete noncompact manifolds with harmonic curvature

Let (Mn, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mn, g) is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mn, g) with positive scalar curvature.

Prescribing the scalar curvature problem on higher-dimensional manifolds

In this paper we consider the problem of existence of conformalmetrics with prescribed scalar curvature on n-dimensionalRiemannian manifolds, $n \geq 5 $. Using precise estimates on thelosses of compactness, we characterize the critical points atinfinity of the associated variational problem and we proveexistence results for curvatures satisfying an assumption ofBahri-Coron type.

Best constants in second-order Sobolev inequalities on compact Riemannian manifolds in the presence of symmetries

Let (M,g) be a smooth compact 3\leq n-dimensional Riemannian manifold, and G a subgroup of the isometry group of (M,g). We establish the best constants in second-order for a Sobolev inequality when the functions are G-invariant.

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = eh dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \({\triangle_{\mu}}\) has a positive lower bound λ1(M) > 0 and the m(m > n)-dimensional Bakry-Emery curvature is bounded from below by \({-\frac{m-1}{m-2}\lambda_1(M)}\), then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.

Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space

We study a geometric problem that originates from theories of nonlinear elasticity: given a non-flat n-dimensional Riemannian manifold with boundary, homeomorphic to a bounded subset of ℝ n , what is the minimum amount of deformation required in order to immerse it in a Euclidean space of the same dimension? The amount of deformation, which in the physical context is an elastic energy, is quantified by an average over a local metric discrepancy. We derive an explicit lower bound for this energy for the case where the scalar curvature of the manifold is non-negative. For n = 2 we generalize the result for surfaces of arbitrary curvature.

ISOMETRIC IMMERSIONS INTO LORENTZIAN PRODUCTS

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n× ℝ1or ℍn× ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

The relative isoperimetric inequality on a conformally parabolic manifold with boundary

For an arbitrary noncompact n-dimensional Riemannian manifold with a boundary of conformally parabolic type it is proved that there exists a conformal change of metric such that a relative isoperimetric inequality of the same form as in the closed n-dimensional Euclidean half-space holds on the manifold with the new metric. This isoperimetric inequality is asymptotically sharp.Bibliography: 6 titles.

Bishop-Gromov type inequality on Ricci limit spaces

In this paper, we study limit spaces of a sequence of n-dimensional complete Riemannian manifolds whose Ricci curvatures have definite lower bound. We will give several measure theoretical properties of such limit spaces.

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n > 7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σi has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface \({{\hat \Sigma}_i \subset \mathbb{R}^n}\), then $$ \int\limits_{\Sigma_i} H d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} $$ where H is the mean curvature of Σi in (Ω, g), \({\hat{H}}\) is the Euclidean mean curvature of \({{\hat \Sigma}_i}\) in \({\mathbb{R}^n}\), and where d σ and \({d {\hat \sigma}}\) denote the respective volume forms. Moreover, equality holds for some boundary component Σi if, and only if, (Ω, g) is isometric to a domain in \({\mathbb{R}^n}\). In the proof, we make use of a foliation of the exterior of the \({\hat \Sigma_i}\)’s in \({\mathbb{R}^n}\) by the \({\frac{H}{R}}\)-flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).

Toric Integrable Geodesic Flows in Odd Dimensions

Let Q be a compact, connected n-dimensional Riemannian manifold, and assume that the geodesic flow is toric integrable. If n 6= 3 is odd, or if π1(Q) is infinite, we show that the cosphere bundle of Q is equivariantly contactomorphic to the cosphere bundle of the torus Tn. As a consequence, Q is homeomorphic to Tn.

On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N, h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T),W 1,n ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ L ∞ L 2 ∩ L 2 H 1 , ▿ P ∈ L 4/3 L 4/3 , and ▿d ∈ L ∞ L 2 ∩ L 2 H 2 ; or (ii) for n = 3, u ∈ L ∞ L 2 ∩ L 2 H 1 ∩ C ([0, T), L n ), P ∈ L /2 L /2 , and ▿d ∈ L 2 L 2 ∩ C ([0, T), L n ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let $${(\mathcal {M},\tilde{g})}$$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation $$\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},$$ where $${\varepsilon}$$ is a small parameter. Let $${{\mathcal {K} \subset \mathcal {M}}}$$ be an (N - 1)-dimensional smooth minimal submanifold that separates $${\mathcal{M}}$$ into two disjoint components. Assume that $${\mathcal{K}}$$ is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that $${{|A_\mathcal {K} |^2 + {\rm {Ric}}_{\tilde g} (v _{\mathcal K}, v_{\mathcal K})}}$$ is positive along $${\mathcal{K}}$$ . Then for each integer m ≥ 2, we establish the existence of a sequence $${\varepsilon = \varepsilon{_j} \rightarrow 0}$$ , and solutions $${u_\varepsilon}$$ with m-transition layers near $${\mathcal{K}}$$ , with mutual distance $${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$$ .